Study Notes on Limits and Continuity

Unit 1: Limits and Continuity

Introduction to Limits

Limits are defined as the value that a function approaches as the variable within the function nears a particular value. It is important to note that while evaluating limits, we focus on the behavior of the function around that value rather than the value itself. This means that we are often concerned with what happens to the function as it approaches a point from either side.

Ways to Find Limits

1. Graphical Analysis: Examine the graph of the function to identify what value the function approaches as the input variable approaches a specific number.

2. Existence of Limit: If the graph approaches two different values for the same input (for example, when approached from the left and right), this indicates that the limit does not exist.

3. Tabular Estimation: Create a table of values that approach the point from either direction and observe the trend in the function's outputs.

4. Algebraic Properties: Utilize analytical methods to compute the limit directly.

Algebraic Manipulation for Finding Limits

Algebraic techniques allow you to manipulate the expressions to find limits more easily, particularly in cases involving polynomials:

• Factoring Technique: One can factor the numerator and denominator, canceling any removable discontinuities. This is especially useful when the denominator evaluates to zero, which would otherwise make the limit undefined.

  • Example: For the limit of the function $( rac{(x+3)(x+2)}{(x+3)(x-3)})$, you can cancel $(x+3)$ if $x <br>eq -3$, thus turning a potential undefined expression into a computable one.

The Squeeze Theorem

The Squeeze Theorem provides a method for determining the limit of a function defined between two others:

• Conditions: If, for all values of $x$ in an interval containing $a$, the inequalities $g(x) \leq f(x) \leq h(x)$ hold and both $g$ and $h$ approach the same limit as $x$ approaches $a$, then:

  • $ext{If } ext{lim}<em>{x o a} g(x) = L ext{ and } ext{lim}</em>{x o a} h(x) = L, ext{ then } ext{lim}_{x o a} f(x) = L.$

Trigonometric Limits

Several fundamental limits involving trigonometric functions as $x$ approaches 0 are established as follows:

• $ext{lim}_{x o 0} rac{ ext{sin}(x)}{x} = 1$

• $ext{lim}_{x o 0} rac{ ext{cos}(x)-1}{x} = 0$

• $ext{lim}_{x o 0} rac{ ext{sin}(ax)}{x} = a$

• $ext{lim}_{x o 0} rac{ ext{sin}(ax)}{ ext{sin}(bx)} = rac{a}{b}$

Types of Discontinuities

Discontinuities in functions can be classified into three main categories:

1. Jump Discontinuity: Occurs when the curve has a break at a certain point, meaning the left- and right-hand limits exist but do not equal each other.

2. Essential (Infinite) Discontinuity: Characterized by the presence of a vertical asymptote within the curve, typically indicating that the limits approach infinity.

3. Removable Discontinuity: This occurs when there is a hole in the curve. It is considered removable because one can redefine the function at that point to fill the gap, allowing for continuity.

Continuity Conditions

To determine whether a function $f(x)$ is continuous at a particular point $x=c$, the following conditions must be satisfied:

1. $f(c)$ exists.

2. The limit $ext{lim}_{x o c} f(x)$ exists.

3. The limit must equal the function value: $ext{lim}_{x o c} f(x) = f(c).$

A function is said to be continuous on an interval if it is continuous at every point within that interval.

Removing Discontinuities

Discontinuities can often be removed by redefining the function to omit the problematic point. This correction is frequently done by factoring out any common roots in both the numerator and denominator, which may help to eliminate expressions that yield undefined points.

Limits and Asymptotes

• Vertical Asymptote: A vertical line where a function cannot cross due to being undefined at that point. This typically correlates with the presence of a limit approaching infinity.

• Horizontal Asymptote: Reflects the end behavior of a function as $x$ approaches infinity or negative infinity. Unlike vertical asymptotes, horizontal asymptotes can potentially be crossed by the function.

Horizontal Asymptote Rules

1. If the highest power of $x$ is in the numerator of a rational expression, the limit as $x$ approaches infinity is infinity, meaning that there is no horizontal asymptote.

2. If the highest power is in the denominator, the limit approaches zero; thus, the horizontal asymptote is the line $y=0$.

3. If the highest powers in the numerator and denominator are the same, the limit equals the coefficient of the highest term in the numerator divided by the coefficient of the highest term in the denominator.

Intermediate Value Theorem (IVT)

The Intermediate Value Theorem states that if a function $f(x)$ is continuous on the closed interval $[a, b]$ and $C$ is any number between $f(a)$ and $f(b)$, then there exists at least one number $d$ in the interval $[a, b]$ such that $f(d) = C$. This theorem is integral to understanding the properties of continuous functions and their behavior across intervals.