Study Guide: Limits and Techniques for Calculus Exams

Limits and Their Evaluation

Introduction

This section covers the foundational concepts of limits in calculus, including techniques for evaluating limits, the use of graphs, and special cases such as piecewise and infinite limits. Mastery of these topics is essential for success in introductory calculus courses and exams.

One-Sided and Two-Sided Limits

• Definition: A limit describes the value that a function approaches as the input approaches a certain point.

• One-sided limits: The value the function approaches from one side (left or right) of a point, denoted as $\lim_{x \to a^-} f(x)$ (from the left) and $\lim_{x \to a^+} f(x)$ (from the right).

• Two-sided limits: The value the function approaches from both sides, denoted as $\lim_{x \to a} f(x)$, exists only if both one-sided limits are equal.

• Graphical Evaluation: Limits can often be estimated or confirmed by analyzing the graph of a function near the point of interest.

• Example: For $f(x) = |x|/x$, $\lim_{x \to 0^-} f(x) = -1$ and $\lim_{x \to 0^+} f(x) = 1$, so $\lim_{x \to 0} f(x)$ does not exist.

Techniques for Evaluating Limits

• Factoring: Simplify the function by factoring and canceling common terms to resolve indeterminate forms such as $0/0$.

• Trigonometric Identities: Use identities to simplify trigonometric expressions. For example, $\sin^2(u) + \cos^2(u) = 1$.

• The Squeeze Theorem: If $g(x) \leq f(x) \leq h(x)$ for all $x$ near $a$, and $\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$, then $\lim_{x \to a} f(x) = L$.

• Example (Factoring): $\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = \lim_{x \to 2} \frac{(x-2)(x+2)}{x-2} = \lim_{x \to 2} (x+2) = 4$.

• Example (Squeeze Theorem): $\lim_{x \to 0} x^2 \sin(1/x) = 0$ because $-x^2 \leq x^2 \sin(1/x) \leq x^2$ and both bounds approach 0 as $x \to 0$.

Piecewise Limits

• Definition: A piecewise function is defined by different expressions over different intervals.

• Analytical Evaluation: To find the limit at a point where the definition changes, evaluate the left- and right-hand limits using the appropriate expressions.

• Example: For $f(x) = \begin{cases} x^2 & x < 1 \\ 2x & x \geq 1 \end{cases}$, $\lim_{x \to 1^-} f(x) = 1$, $\lim_{x \to 1^+} f(x) = 2$; the two-sided limit does not exist at $x=1$.

Infinite Limits and Limits at Infinity

• Infinite Limits: When $f(x)$ increases or decreases without bound as $x$ approaches a certain value, we write $\lim_{x \to a} f(x) = \infty$ or $-\infty$.

• Limits at Infinity: Describes the behavior of $f(x)$ as $x$ approaches $\infty$ or $-\infty$.

• Example: $\lim_{x \to \infty} \frac{1}{x} = 0$.

• Horizontal Asymptotes: If $\lim_{x \to \infty} f(x) = L$, then $y = L$ is a horizontal asymptote.

Summary Table: Techniques for Evaluating Limits

Technique	When to Use	Example
Factoring	Indeterminate forms like $0/0$	$\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$
Trigonometric Identities	Limits involving trig functions	$\lim_{x \to 0} \frac{1 - \cos(x)}{x^2}$
Squeeze Theorem	Functions bounded by two others	$\lim_{x \to 0} x^2 \sin(1/x)$
Graphical Analysis	Visualizing function behavior	Estimating $\lim_{x \to a} f(x)$ from a graph