Calculus I Exam 1 Review: Limits, Continuity, and Asymptotes

Limits and Their Properties

Average Velocity Formula

The average velocity of an object over a time interval [a, b] is calculated using the change in position divided by the change in time:

• Formula: $v_{av} = \frac{s(b) - s(a)}{b - a}$

• Application: Used to estimate instantaneous velocity as the interval shrinks.

Evaluating Limits from a Graph

To evaluate limits from a graph, observe the behavior of the function as x approaches a specific value from both sides. The limit exists if the function approaches the same value from the left and right.

• Left-hand limit: $\lim\limits_{x \to a^-} f(x)$

• Right-hand limit: $\lim\limits_{x \to a^+} f(x)$

• Two-sided limit: $\lim\limits_{x \to a} f(x)$ exists if both one-sided limits are equal.

Limit Laws

The following properties, known as Limit Laws, allow us to compute limits analytically when the limits of individual functions exist:

• Sum Law: $\lim\limits_{x \to a} [f(x) + g(x)] = \lim\limits_{x \to a} f(x) + \lim\limits_{x \to a} g(x)$

• Difference Law: $\lim\limits_{x \to a} [f(x) - g(x)] = \lim\limits_{x \to a} f(x) - \lim\limits_{x \to a} g(x)$

• Constant Multiple Law: $\lim\limits_{x \to a} [c f(x)] = c \lim\limits_{x \to a} f(x)$

• Product Law: $\lim\limits_{x \to a} [f(x)g(x)] = (\lim\limits_{x \to a} f(x))(\lim\limits_{x \to a} g(x))$

• Quotient Law: $\lim\limits_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim\limits_{x \to a} f(x)}{\lim\limits_{x \to a} g(x)}$, provided $\lim\limits_{x \to a} g(x) \neq 0$

• Power Law: $\lim\limits_{x \to a} [f(x)]^n = [\lim\limits_{x \to a} f(x)]^n$

• Root Law: $\lim\limits_{x \to a} [f(x)]^{1/n} = [\lim\limits_{x \to a} f(x)]^{1/n}$, provided $\lim\limits_{x \to a} f(x) > 0$ for even n

Evaluating Limits Analytically

To evaluate limits analytically, apply the limit laws and algebraic manipulation. For example, if $f(x) \to -75$ and $g(x) \to 0$ with $g(x) > 0$ as $x \to 3$, then:

• $\lim\limits_{x \to 3} \frac{f(x)}{g(x)} = -\infty$

Infinite Limits and Asymptotes

Vertical Asymptotes

A vertical asymptote occurs at $x = a$ if the function grows without bound as x approaches a from either side:

• If $\lim\limits_{x \to a^-} f(x) = \pm \infty$ or $\lim\limits_{x \to a^+} f(x) = \pm \infty$, then $x = a$ is a vertical asymptote.

Limits at Infinity and Horizontal Asymptotes

As x approaches infinity, if $f(x)$ approaches a finite value L, then the line $y = L$ is a horizontal asymptote of $f(x)$:

• $\lim\limits_{x \to \infty} f(x) = L$

• $\lim\limits_{x \to -\infty} f(x) = M$

• Both $y = L$ and $y = M$ are horizontal asymptotes if these limits exist.

Infinite Limits at Infinity

If $f(x)$ becomes arbitrarily large as $x$ becomes arbitrarily large, we write:

• $\lim\limits_{x \to \infty} f(x) = \infty$

• Similarly, $\lim\limits_{x \to \infty} f(x) = -\infty$, $\lim\limits_{x \to -\infty} f(x) = \infty$, and $\lim\limits_{x \to -\infty} f(x) = -\infty$ are defined.

Limits at Infinity of Powers and Polynomials

For polynomials and powers, the behavior as $x \to \infty$ or $x \to -\infty$ depends on the degree and leading coefficient:

• If $n$ is even: $\lim\limits_{x \to \infty} x^n = \infty$, $\lim\limits_{x \to -\infty} x^n = \infty$

• If $n$ is odd: $\lim\limits_{x \to \infty} x^n = \infty$, $\lim\limits_{x \to -\infty} x^n = -\infty$

• $\lim\limits_{x \to \infty} x^{-n} = 0$, $\lim\limits_{x \to -\infty} x^{-n} = 0$

• For a polynomial $p(x)$, $\lim\limits_{x \to \infty} p(x) = \lim\limits_{x \to \infty} a_n x^n = \pm \infty$, depending on the sign of $a_n$

Continuity and Discontinuity

Continuity at a Point

A function $f(x)$ is continuous at $x = a$ if:

• $f(a)$ is defined

• $\lim\limits_{x \to a} f(x)$ exists

• $\lim\limits_{x \to a} f(x) = f(a)$

Points of Discontinuity

A point of discontinuity occurs where a function is not continuous. This can happen if the function is not defined, the limit does not exist, or the limit does not equal the function value at that point.

Summary Table: Types of Asymptotes

Type	Definition	Equation
Vertical Asymptote	Function approaches infinity as x approaches a	x = a
Horizontal Asymptote	Function approaches a finite value as x approaches infinity	y = L