Comprehensive Study Notes on Real Valued Functions and Calculus Basics

Real Valued Function of Real Variables

Definition of Set and Function: Let $\mathbb{R}$ denote the set of all real numbers. Let $S$ be any subset of $\mathbb{R}$ . The function $f$ is defined as $f: S \rightarrow \mathbb{R}$ , where $\mathbb{R} = (-\infty, \infty)$ .
Classification of Real Valued Functions: A system of functions where a real number is input and it generates a real number is called a real valued function of real variables ( $f$ is a function that maps each element of $S$ of $\mathbb{R}$ ).
Illustrative Example 1: Consider $f(x) = x+1$ .
- At $x=0$ , $f(0) = 0+1 = 1$ .
- At $x=1$ , $f(1) = 1+1 = 2$ .
Illustrative Example 2: Consider $f(x) = x+2$ .
- At $x=1$ , $f(1) = 1+2 = 3$ .
- At $x=-7$ , $f(-7) = -7+2 = -5$ .
- At $x=0$ , $f(0) = 0+2 = 2$ .
- In this system, each time a real number is substituted for $x$ , a corresponding real number is obtained as the value of $f(x)$ . This confirms that $f(x)$ is a real valued function of real variables.

Specific Real Valued Functions

The Constant Function:
- $f$ is a constant function if $f(x) = k$ for all $x \in S$ , where $k$ is a constant and $S$ is the domain of $f$ .
- Example: $f(x) = 2$ . Any input results in 2: $f(\frac{1}{2}) = 2$ , $f(10) = 2$ , $f(0) = 2$ , $f(30) = 2$ .
- The value of $f(x)$ remains constant even though $x$ may be changing.
The Identity Function:
- Let $f$ be a function such that $f(x) = x$ for all $x$ .
- Whatever you input is what you generate as output.
- Example: $f(3) = 3$ .
- In another form, it is written as $Id(x) = x$ , where $Id$ represents the identity function.
The Signum Function ( $Sgn$ ):
- Otherwise called the "Sign Function," it is defined by: $Sgn(x) = \begin{cases} 1 & \text{for } x > 0 \\ 0 & \text{for } x = 0 \\ -1 & \text{for } x < 0 \end{cases}$
- Case 1 (x > 0): $Sgn(10) = 1$ because 10 > 0. $Sgn(3) = 1$ . $Sgn(4) = 1$ . $Sgn(0.000001) = 1$ .
- Case 2 (x = 0): $Sgn(x) = 0$ for $x = 0$ . Example: $Sgn(\pi - \pi) = 0$ .
- Case 3 (x < 0): $Sgn(x) = -1$ . Example: $Sgn(-10) = -1$ since -10 < 0. $Sgn(-100) = -1$ . $Sgn(-0.001) = -1$ .
The Absolute or Modulus Function:
- Shortened as "mod," it is defined by: $mod(x) \text{ or } |x| = \begin{cases} x & \text{for } x > 0 \\ 0 & \text{for } x = 0 \\ -x & \text{for } x < 0 \end{cases}$
- It converts all positive values to positive and negative values to positive. It is used to remove negativity. Examples: $|-10| = 10$ , $|-20| = 20$ , $|10| = 10$ , $|0| = 0$ .
The Heaviside Function:
- Denoted by $H$ and defined by: $H(x) = \begin{cases} 1 & \text{for } x > 0 \\ 0 & \text{for } x \le 0 \end{cases}$
- Case 1 (x > 0): $H(1) = 1$ , $H(\pi) = 1$ , $H(0.0001) = 1$ .
- Case 2 (x ≤ 0): $H(-10) = 0$ since -10 < 0. $H(0) = 0$ because $0 \le 0$ . $H(-\pi) = 0$ .
The Greatest Integer Function:
- Denoted by $[x]$ and defined by $[x] = b$ , where $b$ is the greatest integer $\le x$ .
- Calculation Rule: To calculate the value, always look to the left on a number line and notice the next whole number.
- For Positive Numbers: It is the whole number before the decimal. Example: $[7.35] = 7$ , $[10] = 10$ , $[0.5] = 0$ .
- For Negative Numbers: Add one magnitude-wise (look left). Example: $[-3.72] = -4$ , $[-6.282] = -7$ , $[-7] = -7$ .

Algebraic Classifications of Functions

Linear Function: Defined as $f(x) = ax + b$ .
Polynomial Function: Defined as $f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_2 x^2 + a_1 x^1 + a_0 x^0$ , where $a_n$ are constants.
- Degree 0: $f(x) = 10$ .
- Degree 2 (Quadratic): $f(x) = 2x^2 + 3x + 1$ .
- Degree 3 (Cubic): $f(x) = 3x^3 + 2x^2 + 2x + 4$ .
- Degree 4 (Quartic): $f(x) = 3x^4 + 4x^2 + 5x$ .
Rational Function: Defined as $f(x) = \frac{p(x)}{q(x)}$ , where $q(x) \ne 0$ and both $p(x)$ and $q(x)$ are polynomials. Example: $f(x) = \frac{2x+3}{x^2+3x+1}$ .

Variables and Domain/Range Concepts

Variables:
- $x$ is the Input Variable (Independent Variable).
- $y$ is the Output Variable (Dependent Variable).
Domain: The set of all values of $x$ for which the function is defined and can calculate corresponding values for $f(x)$ .
Range: The set of corresponding values of $f(x)$ .
Example Calculation: For $f(x) = 3x^2 + 2$ :
- If Domain = $\{1, 2, 3\}$
- $f(1) = 3(1^2) + 2 = 5$
- $f(2) = 3(2^2) + 2 = 14$
- $f(3) = 3(3^2) + 2 = 29$
- Range = $\{5, 14, 29\}$ .
- Co-domain = All whole numbers $\ge 5$ .

The Inverse Function

Definition: To find the inverse function $f^{-1}(x)$ .
Procedure Steps:
1. Set $f(x) = y$ .
2. Make $x$ the subject of the formula.
3. Replace $y$ by $x$ in the resulting expression.
Inverse Example 1: Given $f(x) = 3x + 4$ .
- Set $y = 3x + 4$ .
- $3x = y - 4 \rightarrow x = \frac{y-4}{3}$ .
- $f^{-1}(x) = \frac{x-4}{3}$ .
Inverse Example 2: Given $f(x) = \sqrt{\frac{1}{2}x + 1}$ .
- $y = \sqrt{\frac{1}{2}x + 1} \rightarrow y^2 = \frac{1}{2}x + 1$ .
- $y^2 - 1 = \frac{x}{2} \rightarrow x = 2(y^2 - 1)$ .
- $f^{-1}(x) = 2(x^2 - 1)$ .
Inverse Example 3: Given $f(x) = \ln(x+4)$ .
- $y = \ln(x+4)$ . Take exponential of both sides.
- $e^y = x+4 \rightarrow x = e^y - 4$ .
- $f^{-1}(x) = e^x - 4$ .
Inverse Example 4: Given $f(x) = e^{3x} - 4$ .
- $y = e^{3x} - 4 \rightarrow y+4 = e^{3x}$ .
- Take $\ln$ on both sides: $\ln(y+4) = 3x \rightarrow x = \frac{\ln(y+4)}{3}$ .
- $f^{-1}(x) = \frac{\ln(x+4)}{3}$ .

Composite Functions

Definition: If given $f(x) = 2x+1$ , $g(x) = 3x+2$ , and $h(x) = x+1$ , a composite function like $f \circ g \circ h(x)$ means putting one function's output as the input of the next.
Example 1: Find $f \circ g(x)$ .
- $f(g(x)) = f(3x+2) = 2(3x+2) + 1 = 6x+4+1 = 6x+11$ .
Example 2: Find $g \circ f \circ h(x)$ at $x=2$ .
- $h(2) = 2+1 = 3$ .
- $f(3) = 2(3)+1 = 7$ .
- $g(7) = 3(7)+2 = 23$ .
Example 3: Find $f \circ f(x)$ where $f(x) = x^2+1$ .
- $f(f(x)) = f(x^2+1) = (x^2+1)^2 + 1 = x^4 + 2x^2 + 1 + 1 = x^4 + 2x^2 + 2$ .

Finding the Domain

Techniques for Finding Domain:
1. For Polynomial Functions: The domain is always all real numbers $\mathbb{R} = (-\infty, \infty)$ . Example: $f(x) = 2x^2 - 4x + 5$ , $\text{dom} (f) = \mathbb{R}$ .
2. For Rational Functions: We avoid any value of $x$ that makes the denominator zero.
  - Example: $f(x) = \frac{2x-4}{x+1}$ . Set $x+1=0 \rightarrow x=-1$ . Undefined at $x=-1$ . $\text{dom} (f) = \mathbb{R} \setminus \{-1\}$ or $(-\infty, -1) \cup (-1, \infty)$ .
  - Example: $f(x) = \frac{x}{x^2-x-6}$ . Set $x^2-x-6=0 \rightarrow (x-3)(x+2)=0$ . Undefined at $x=3, -2$ . $\text{dom} (f) = \mathbb{R} \setminus \{-2, 3\}$ .
3. For Radical (Square Root) Functions:
  - The expression under the root ( $x$ ) must be $\ge 0$ if the root index $n$ is even.
  - If $n$ is odd, the domain is the domain of the underlying polynomial ( $\mathbb{R}$ ).
  - Example: $f(x) = \sqrt{x^2+x-6}$ . Require $x^2+x-6 \ge 0 \rightarrow (x-2)(x+3) \ge 0$ . $\text{dom} (f) = (-\infty, -3] \cup [2, \infty)$ .
  - Example (Odd root): $f(x) = \sqrt[3]{x^2-x-6}$ . $\text{dom} (f) = \mathbb{R}$ .

Limits of a Function

Mathematical Definition: The limit of $f(x)$ as $x$ approaches a point $c$ is $L$ , written $\lim_{x \to c} f(x) = L$ . This means there exists some number \epsilon > 0 such that |f(x) - L| < \epsilon for all $x$ such that |x - c| < \delta, where \delta > 0.
Properties of Limits:
1. Constant: $\lim_{x \to c} k = k$ for all $c \in \mathbb{R}$ .
2. Identity: $\lim_{x \to c} x = c$ .
3. Constant Multiple: $\lim_{x \to c} k f(x) = k \lim_{x \to c} f(x)$ .
4. Sum/Difference: $\lim_{x \to c} [f(x) \pm g(x)] = \lim_{x \to c} f(x) \pm \lim_{x \to c} g(x)$ .
5. Rational Power: $\lim_{x \to c} [f(x)]^n = [\lim_{x \to c} f(x)]^n$ , provided \lim_{x \to c} f(x) > 0 if $n$ is even.
Basic Limit Evaluations:
- $\lim_{x \to 2} (2x^2 - 3x + 1) = 2(2^2) - 3(2) + 1 = 8 - 6 + 1 = 3$ .
- $\lim_{x \to -1} \sqrt[3]{5x^3 - 3} = \sqrt[3]{5(-1)^3 - 3} = \sqrt[3]{-8} = -2$ .
Handling Indeterminate Forms (e.g., 0/0):
- Method I (Simplification/Factoring): Factor the numerator and denominator to cancel terms.
  - Example: $\lim_{x \to -2} \frac{x^2+x-2}{x^2+2x}$ . Factor: $\frac{(x-1)(x+2)}{x(x+2)} = \frac{x-1}{x}$ . Limit is $\frac{-2-1}{-2} = \frac{-3}{-2} = 1.5$ .
- Method II (L'Hopital's Rule): Differentiate the numerator and denominator separately and then substitute the point.
Limit as $x \to \infty$ :
- Divide the numerator and denominator by the highest power of $x$ present.
- Shortcut:
  1. If numerator and denominator have the same highest power, the limit is the ratio of their coefficients. Example: $\lim_{x \to \infty} \frac{2x^2+3x}{4x^2+5x+1} = \frac{2}{4} = 0.5$ .
  2. If the highest power is in the denominator, the limit is 0. Example: $\lim_{x \to \infty} \frac{5x^3-7x+3}{x^5-4x^2+4} = 0$ .
  3. If the highest power is in the numerator, the limit is $\infty$ (undefined).

Review Questions & Discussion

Question 1: Simplify $[-5.76] + H[5.76] + Sgn(5.76)$ .
- $-5.76$ is between $-6$ and $-5$ , so $[-5.76] = -6$ .
- $H(5.76) = 1$ (since 5.76 > 0).
- $Sgn(5.76) = 1$ .
- Result: $-6 + 1 + 1 = -4$ .
Question 2: Find $f(\frac{3}{4})$ if $f(x) = \frac{x - [x]}{x}$ .
- $f(\frac{3}{4}) = \frac{\frac{3}{4} - [\frac{3}{4}]}{\frac{3}{4}} = \frac{\frac{3}{4} - 0}{\frac{3}{4}} = \frac{\frac{3}{4}}{\frac{3}{4}} = 1.\n\n* **Question 3:** Evaluate $[Sgn(H(-3.4))]$.\n * H(-3.4) = 0 $since$ -3.4 < 0.\n * Sgn(0) = 0.\n * [0] = 0.\n\n* **Question 4:** Find range for [x+2] = 0.\n * [x+2] = 0 $implies$ 0 \le x+2 < 1.\n * Subtract 2: -2 \le x < -1.\n\n* **Question 5:** What is the range for which [x-4] = 2?\n * 2 \le x-4 < 3.\n * Add 4: 6 \le x < 7.", "title": "Real Valued Functions and Calculus Study Guide"} Ready for use. \n\n\n```json\n{\n "note": "# Real Valued Function of Real Variables\n\n* **Definition of Set and Function:** Let \mathbb{R} $denote the set of all real numbers. Let$ S $be any subset of$ \mathbb{R} $. The function$ f $is defined as$ f: S \rightarrow \mathbb{R} $, where$ \mathbb{R} = (-\infty, \infty).\n\n* **Classification of Real Valued Functions:** A system of functions where a real number is input and it generates a real number is called a real valued function of real variables (f $is a function that maps each element of$ S $of$ \mathbb{R}).\n\n* **Illustrative Example 1:** Consider f(x) = x+1.\n * At x=0 $,$ f(0) = 0+1 = 1.\n * At x=1 $,$ f(1) = 1+1 = 2.\n\n* **Illustrative Example 2:** Consider f(x) = x+2.\n * At x=1 $,$ f(1) = 1+2 = 3.\n * At x=-7 $,$ f(-7) = -7+2 = -5.\n * At x=0 $,$ f(0) = 0+2 = 2.\n * In this system, each time a real number is substituted for x $, a corresponding real number is obtained as the value of$ f(x) $. This confirms that$ f(x) is a real valued function of real variables.\n\n# Specific Real Valued Functions\n\n* **The Constant Function:**\n * f $is a constant function if$ f(x) = k $for all$ x \in S $, where$ k $is a constant and$ S $is the domain of$ f.\n * Example: f(x) = 2 $. Any input results in 2:$ f(\frac{1}{2}) = 2 $,$ f(10) = 2 $,$ f(0) = 2 $,$ f(30) = 2.\n * The value of f(x) $remains constant even though$ x may be changing.\n\n* **The Identity Function:**\n * Let f $be a function such that$ f(x) = x $for all$ x. \n * Whatever you input is what you generate as output. \n * Example: f(3) = 3. \n * In another form, it is written as Id(x) = x $, where$ Id represents the identity function.\n\n* **The Signum Function (Sgn):**\n * Otherwise called the \"Sign Function,\" it is defined by:\n Sgn(x) = \begin{cases} 1 & \text{for } x > 0 \\ 0 & \text{for } x = 0 \\ -1 & \text{for } x < 0 \end{cases}\n * **Case 1 (x > 0):** Sgn(10) = 1 $because$ 10 > 0 $.$ Sgn(3) = 1 $.$ Sgn(4) = 1 $.$ Sgn(0.000001) = 1.\n * **Case 2 (x = 0):** Sgn(x) = 0 $for$ x = 0 $. Example:$ Sgn(\pi - \pi) = 0.\n * **Case 3 (x < 0):** Sgn(x) = -1 $. Example:$ Sgn(-10) = -1 $since$ -10 < 0 $.$ Sgn(-100) = -1 $.$ Sgn(-0.001) = -1.\n\n* **The Absolute or Modulus Function:**\n * Shortened as \"mod,\" it is defined by:\n mod(x) \text{ or } |x| = \begin{cases} x & \text{for } x > 0 \\ 0 & \text{for } x = 0 \\ -x & \text{for } x < 0 \end{cases}\n * It converts all positive values to positive and negative values to positive. It is used to remove negativity. Examples: |-10| = 10 $,$ |-20| = 20 $,$ |10| = 10 $,$ |0| = 0.\n\n* **The Heaviside Function:**\n * Denoted by H and defined by:\n H(x) = \begin{cases} 1 & \text{for } x > 0 \\ 0 & \text{for } x \le 0 \end{cases}\n * **Case 1 (x > 0):** H(1) = 1 $,$ H(\pi) = 1 $,$ H(0.0001) = 1.\n * **Case 2 (x \\le 0):** H(-10) = 0 $since$ -10 < 0 $.$ H(0) = 0 $because$ 0 \le 0 $.$ H(-\pi) = 0.\n\n* **The Greatest Integer Function:**\n * Denoted by [x] $and defined by$ [x] = b $, where$ b $is the greatest integer$ \le x.\n * **Calculation Rule:** To calculate the value, always look to the left on a number line and notice the next whole number.\n * **For Positive Numbers:** It is the whole number before the decimal. Example: [7.35] = 7 $,$ [10] = 10 $,$ [0.5] = 0.\n * **For Negative Numbers:** Add one magnitude-wise (look left). Example: [-3.72] = -4 $,$ [-6.282] = -7 $,$ [-7] = -7.\n\n# Algebraic Classifications of Functions\n\n* **Linear Function:** Defined as f(x) = ax + b.\n\n* **Polynomial Function:** Defined as f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_2 x^2 + a_1 x^1 + a_0 x^0 $, where$ a_n are constants.\n * **Degree 0:** f(x) = 10.\n * **Degree 2 (Quadratic):** f(x) = 2x^2 + 3x + 1. \n * **Degree 3 (Cubic):** f(x) = 3x^3 + 2x^2 + 2x + 4.\n * **Degree 4 (Quartic):** f(x) = 3x^4 + 4x^2 + 5x.\n\n* **Rational Function:** Defined as f(x) = \frac{p(x)}{q(x)} $, where$ q(x) \ne 0 $and both$ p(x) $and$ q(x) $are polynomials. Example:$ f(x) = \frac{2x+3}{x^2+3x+1}.\n\n# Variables and Domain/Range Concepts\n\n* **Variables:**\n * x is the Input Variable (Independent Variable).\n * y is the Output Variable (Dependent Variable).\n\n* **Domain:** The set of all values of x $for which the function is defined and can calculate corresponding values for$ f(x). \n\n* **Range:** The set of corresponding values of f(x).\n\n* **Example Calculation:** For f(x) = 3x^2 + 2:\n * If Domain = \{1, 2, 3\}\n * f(1) = 3(1^2) + 2 = 5\n * f(2) = 3(2^2) + 2 = 14\n * f(3) = 3(3^2) + 2 = 29\n * Range = \{5, 14, 29\}. Domain values are input, calculated values form the range.\n\n# The Inverse Function\n\n* **Definition:** To find the inverse function f^{-1}(x).\n\n* **Procedure Steps:**\n 1. Set f(x) = y.\n 2. Make x the subject of the formula.\n 3. Replace y $by$ x in the resulting expression.\n\n* **Inverse Example 1:** Given f(x) = 3x + 4.\n * Set y = 3x + 4.\n * 3x = y - 4 \rightarrow x = \frac{y-4}{3}.\n * f^{-1}(x) = \frac{x-4}{3}.\n\n* **Inverse Example 2:** Given f(x) = \sqrt{\frac{1}{2}x + 1}.\n * y = \sqrt{\frac{1}{2}x + 1} \rightarrow y^2 = \frac{1}{2}x + 1.\n * y^2 - 1 = \frac{x}{2} \rightarrow x = 2(y^2 - 1).\n * f^{-1}(x) = 2(x^2 - 1).\n\n* **Inverse Example 3:** Given f(x) = \ln(x+4).\n * y = \ln(x+4). Take exponential of both sides.\n * e^y = x+4 \rightarrow x = e^y - 4.\n * f^{-1}(x) = e^x - 4.\n\n* **Inverse Example 4:** Given f(x) = e^{3x} - 4.\n * y = e^{3x} - 4 \rightarrow y+4 = e^{3x}.\n * Take \ln $on both sides:$ \ln(y+4) = 3x \rightarrow x = \frac{\ln(y+4)}{3}.\n * f^{-1}(x) = \frac{\ln(x+4)}{3}.\n\n# Composite Functions\n\n* **Definition:** If given f(x) = 2x+1 $,$ g(x) = 3x+2 $, and$ h(x) = x+1 $, a composite function like$ f \circ g \circ h(x) means evaluating one function's output as the input of the next functions in the sequence.\n\n* **Example 1:** Find f \circ g(x).\n * f(g(x)) = f(3x+2) = 2(3x+2) + 1 = 6x+4+1 = 6x+5.\n\n* **Example 2:** Find g \circ f \circ h(x) $at$ x=2.\n * Step 1: Evaluate h(2) = 2+1 = 3.\n * Step 2: Evaluate f(3) = 2(3)+1 = 7.\n * Step 3: Evaluate g(7) = 3(7)+2 = 23.\n\n* **Example 3:** Find f \circ f(x) $where$ f(x) = x^2+1.\n * f(f(x)) = f(x^2+1) = (x^2+1)^2 + 1 = x^4 + 2x^2 + 1 + 1 = x^4 + 2x^2 + 2.\n\n# Finding the Domain\n\n* **Definition:** The domain of a function is the set of values at which the function is defined. Shortened as dom(f).\n\n* **Techniques for Finding Domain:**\n 1. **For Polynomial Functions:** Every polynomial function has a domain of all real numbers \mathbb{R} = (-\infty, \infty).\n 2. **For Rational Functions:** Avoid values of x that make the denominator zero. \n * Example 1: f(x) = \frac{2x-4}{x+1} $. Set$ x+1=0 \rightarrow x=-1 $. Function is undefined at$ x=-1 $.$ \text{dom} (f) = \mathbb{R} \setminus \{-1\} $or$ (-\infty, -1) \cup (-1, \infty).\n * Example 2: f(x) = \frac{x}{x^2-x-6} $. Factor the denominator:$ (x-3)(x+2) = 0 $. Undefined at$ x=3 $and$ x=-2 $.$ \text{dom} (f) = \mathbb{R} \setminus \{-2, 3\}.\n 3. **For Radical (Square Root) Functions:**\n * If the index n $is even, the expression under the root must be greater than or equal to zero ($ \ge 0).\n * If n $is odd, the domain is the same as the polynomial underneath ($ \mathbb{R}).\n * Example (Even Root): f(x) = \sqrt{x^2+x-6} $. Set$ x^2+x-6 \ge 0 \rightarrow (x-2)(x+3) \ge 0 $.$ \text{dom} (f) = (-\infty, -3] \cup [2, \infty).\n * Example (Odd Root): f(x) = \sqrt[3]{x^2-x-6} $. Index is odd, so$ \text{dom} (f) = \mathbb{R}.\n\n# Range and Inverse Relation\n\n* **Key Principle:** The range of a function f $is the same as the domain of its inverse$ f^{-1}.\n* **Finding the Range:** To find the range of f(x), find the inverse function and then determine its domain.\n* **Example:** Find range of f(x) = \frac{3}{x-2}.\n * Inverse: y = \frac{3}{x-2} \rightarrow x-2 = \frac{3}{y} \rightarrow x = \frac{3}{y} + 2 = \frac{3+2y}{y}.\n * f^{-1}(x) = \frac{3+2x}{x} $. The domain of this inverse is$ \mathbb{R} \setminus \{0\}.\n * Range of f(x) = \mathbb{R} \setminus \{0\}.\n\n# Limit of a Function\n\n* **Mathematical Definition:** The limit of f(x) $as$ x $approaches a point$ c $is$ L $, written$ \lim_{x \to c} f(x) = L $. Formally, there exists some number$ \epsilon > 0 $such that$ |f(x) - L| < \epsilon $for all$ x $where$ 0 < |x - c| < \delta.\n\n* **Properties of Limits:**\n 1. **Constant Function:** \lim_{x \to c} k = k $for any constant$ k.\n 2. **Identity Function:** \lim_{x \to c} x = c.\n 3. **Constant Multiple:** \lim_{x \to c} k f(x) = k \lim_{x \to c} f(x).\n 4. **Sum/Difference:** \lim_{x \to c} [f(x) \pm g(x)] = \lim_{x \to c} f(x) \pm \lim_{x \to c} g(x).\n 5. **Rational Power:** \lim_{x \to c} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to c} f(x)} $, provided the root is defined for even$ n.\n\n* **Techniques for Indeterminate Forms (0/0):**\n 1. **Simplification/Factoring:** Factorize terms and cancel common factors.\n * Example: \lim_{x \to -2} \frac{x^2+x-2}{x^2+2x} = \lim_{x \to -2} \frac{(x-1)(x+2)}{x(x+2)} = \lim_{x \to -2} \frac{x-1}{x} = \frac{-3}{-2} = 1.5.\n 2. **L'Hopital's Rule:** Differentiate the numerator and denominator separately until the indeterminacy is removed.\n\n* **Limits at Infinity (x → ∞):**\n * Divide every term by the highest power of x found in the denominator.\n * **Shortcut Rules:**\n 1. Highest power in numerator and denominator are same: The limit is the ratio of their coefficients.\n 2. Highest power is in the denominator: The limit is 0. Examples: \lim_{x \to \infty} \frac{1}{x} = 0 $,$ \lim_{x \to \infty} \frac{1}{x^2} = 0.\n 3. Highest power is in the numerator: The limit is infinity (undefined).\n * Example: \lim_{x \to \infty} \frac{2x^2+3x}{4x^2+5x} = \frac{2}{4} = 0.5$$.", "title": "Comprehensive Study Notes on Real Valued Functions and Calculus Basics" } ```