MATH 1160: Elementary Applied Calculus — Rates of Change and Limits

What is the formal statement of the Existence Theorem for limits?

A two-sided limit exists if and only if both one-sided limits exist and are equal: $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L \iff \lim_{x \to a} f(x) = L$

Given $f(x) = \frac{|x|}{x}$, what are the left-hand and right-hand limits as $x \to 0$?

The left-hand limit is $\lim_{x \to 0^-} \frac{|x|}{x} = -1$ and the right-hand limit is $\lim_{x \to 0^+} \frac{|x|}{x} = 1$. Since they are not equal, $\lim_{x \to 0} \frac{|x|}{x} = \text{DNE}$.

For a piecewise function $g(x)$ where $g(x) = x + 1$ for $x \neq 1$ and $g(1) = 5$, what is $\lim_{x \to 1} g(x)$?

$\lim_{x \to 1} g(x) = 2$. The value of the limit depends strictly on inputs near $x = 1$, not the value defined at $x = 1$.

What behavior occurs for $f(x) = \frac{1}{x}$ as $x$ approaches $0$ from the left and right?

As $x \to 0^-$, $f(x) \to -\infty$. As $x \to 0^+$, $f(x) \to \infty$. Therefore, $\lim_{x \to 0} \frac{1}{x} = \text{DNE}$.

What are the values of $\lim_{x \to \infty} \frac{1}{x}$ and $\lim_{x \to -\infty} \frac{1}{x}$?

$\lim_{x \to \infty} \frac{1}{x} = 0$ and $\lim_{x \to -\infty} \frac{1}{x} = 0$, meaning the graph approaches the horizontal asymptote $y = 0$.

Does evaluating $\lim_{x \to a} f(x)$ require $f(a)$ to exist?

No, evaluating a limit only considers the behavior of $f(x)$ as $x$ approaches $a$, not at $x = a$. The function $f(x)$ does not need to be defined at $x = a$.