Detailed Study Notes on Limits of Functions

LIMIT OF A FUNCTION

Introduction to Limits

• The concept of a “limit” is a fundamental building block for all calculus concepts.
• In this section, the study is largely informal to develop an intuition for the basic ideas.

Definition of Limit

• If the values of f(x) can be made arbitrarily close to L by taking values of x sufficiently close to a (but not equal to a), we express this as:
  • $\lim_{x \to a} f(x) = L$
  • This is read as "the limit of f(x) as x approaches a is L" or "f(x) approaches L as x approaches a."

Example 1: Basic Application of Limits

• To understand how a function behaves as its independent variable approaches a specific value, consider:
  • Function: $f(x) = x^2 - x + 1$
  • We analyze its behavior for x-values close to 2.
  • From the graph and numerical evidence, it is observed that:
  • $\lim_{x \to 2} f(x) = 3$, as the values of f(x) approach 3 for both sides of x = 2.

Example 2: Conjecturing Values of Limits

• Investigate the limit:
  • $\lim_{x \to 1} \frac{x - 1}{\sqrt{x - 1}}$
• Although the function is undefined at x = 1, this does not affect the limit.
• Sample x-values approaching 1 from both sides yield values for f(x) that converge to 2:
  • $\lim_{x \to 1} \frac{x - 1}{\sqrt{x - 1}} = 2$

Numerical Evidence Detailed

• Left-side limits approaching 1:
  • \begin{align} x & : 0.99, 0.999, 0.9999, 0.99999, \ f(x) & : 1.994987, 1.999500, 1.999950, 1.999995 \end{align}
• Right-side limits approaching 1:
  • \begin{align} x & : 1.00001, 1.0001, 1.001, 1.01 \ f(x) & : 2.000005, 2.000050, 2.000500, 2.004988 \end{align}

Limit Problem: Estimating Values

Limit to Evaluate

• $\lim_{x \to 2} \frac{x^2 + 4x - 12}{x^2 - 2x}$
• Table of values for the left ($x -$) and right ($x +$) limits:
  • \begin{array}{|c|c|c|c|}
    \hline
    x & f(x+) & x & f(x-) \
    \hline
    2.5 & 3.4 & 1.5 & 5 \
    2.1 & 3.857142857 & 1.9 & 4.157894737 \
    2.01 & 3.985074627 & 1.99 & 4.015075377 \
    2.0001 & 3.999850007 & 1.9999 & 4.000150008 \
    2.00001 & 3.999985000 & 1.99999 & 4.000015000 \
    \hline
    \end{array}
• Conclusion: The limit approaches 4 as x approaches 2:
  • $\lim_{x \to 2} \frac{x^2 + 4x - 12}{x^2 - 2x} = 4$

One-Sided Limits

• Definitions:
  • Right-Hand Limit: $\lim_{x \to a^+} f(x) = L$, indicating that we approach L from the right (x > a).
  • Left-Hand Limit: $\lim_{x \to a^-} f(x) = L$, indicating that we approach L from the left (x < a).

Example 2: Evaluating One-Sided Limits

• Consider:
  • $H(t) = \begin{cases} 0 &amp; \text{if } t < 0 \\n1 & \text{if } t > 0 \end{cases}$
• We find:
  • $\lim_{t \to 0^+} H(t) = 1$ (from the right)
  • $\lim_{t \to 0^-} H(t) = 0$ (from the left)
• Conclusion: Since the one-sided limits yield different results, $\lim_{t \to 0} H(t)$ does not exist.

Theorems and Properties

Theorem on Existence of Limit

• Given a function $f(x)$, if:
  • $\lim<em>{x \to a^+} f(x) = \lim</em>{x \to a^-} f(x) = L$ then:
  • The normal limit exists: $\lim_{x \to a} f(x) = L$

Properties of Limits

• Assuming $\lim<em>{x \to a} f(x)$ and $\lim</em>{x \to a} g(x)$ exist:
  • 1. $\lim<em>{x \to a} [c f(x)] = c \cdot \lim</em>{x \to a} f(x)$
  • 2. $\lim<em>{x \to a} [f(x) \pm g(x)] = \lim</em>{x \to a} f(x) \pm \lim_{x \to a} g(x)$
  • 3. $\lim<em>{x \to a} [f(x) \cdot g(x)] = \lim</em>{x \to a} f(x) \cdot \lim_{x \to a} g(x)$
  • 4. $\lim<em>{x \to a} \left[ \frac{f(x)}{g(x)} \right] = \frac{\lim</em>{x \to a} f(x)}{\lim<em>{x \to a} g(x)}$, given $\lim</em>{x \to a} g(x) \neq 0$

Theorems Related to Squeeze Theorem

• Squeeze Theorem:
  • If $f(x) \leq g(x) \leq h(x)$ when $x$ approaches $a$ and:
  • $\lim_{x \to a} f(x) = L$
  • $\lim_{x \to a} h(x) = L$
  • Then: $\lim_{x \to a} g(x) = L$

Example Applying Squeeze Theorem

• To show: $\lim_{x \to 0} x^2 \sin(\frac{1}{x}) = 0$
• Using the property which indicates that:
  • Since $-1 \leq \sin(\frac{1}{x}) \leq 1$, it follows:
  • $-x^2 \leq x^2 \sin(\frac{1}{x}) \leq x^2$
• Thus, applying Squeeze Theorem:
  • $\lim_{x \to 0} x^2 \sin(\frac{1}{x}) = 0$

Continuity

Definition of Continuity

• A function $f(x)$ is continuous at $x = a$ if:
  1. $f(a)$ is defined (a is in the domain of f)
  2. $\lim_{x \to a} f(x)$ exists
  3. $\lim_{x \to a} f(x) = f(a)$

Example of Continuity

• Determine continuity at:
  • $x = -2$: Not continuous (jump discontinuity)
  • $x = 0$: Continuous (both limit and function value are equal)
  • $x = 3$: Not continuous (removable discontinuity)

General Continuity Theorems

• Any polynomial function is continuous everywhere.
• Any rational function is continuous wherever it is defined, that is, it is continuous on its domain.

Trials on Continuity and Limits

• Identify non-continuity in given functions:
  1. $h(t) = \frac{4t + 10}{t^2 - 2t - 15}$
  2. f(x) = \begin{cases} \frac{x^2 - x - 2}{x - 2} & \text{if } x \neq 2 \
     6 & \text{if } x = 2 \end{cases}

Conclusion

• An understanding of limits is vital in calculus, as they form the basis for analyzing continuity, derivatives, and integrals.