Question

Given the function f(x) = alt="\frac{|x|}{x}" align="absmiddle" class="latex-formula"> , find $\lim_{x \to \00^-} f(x) and \lim_{x \to \00^+} f(x) and \lim_{x \to \00} f(x)$

Answer 1

Answer:

$\displaystyle \lim_{x \to 0^-} f(x) = -1 \\ \lim_{x \to 0^+} f(x) = 1 \\ \lim_{x \to 0} f(x) = DNE$

General Formulas and Concepts:

Algebra I

• Graphing Functions
• Function Notation

Calculus

Limits

• Evaluating Limits Graphically
• If $\displaystyle \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$, then $\displaystyle \lim_{x \to a} f(x)$ exists

Step-by-step explanation:

Step 1: Define

$\displaystyle f(x) = \frac{|x|}{x}$

Step 2: Graph/Evaluate

Graph the function so we can evaluate the given limits. See attachment.

We see from the function that when we approach 0 from the left, we will get a -1.

∴ $\displaystyle \lim_{x \to 0^-} f(x) = -1$

We see from the function that when we approach 0 from the right, we will get a 1.

∴ $\displaystyle \lim_{x \to 0^+} f(x) = 1$

Since the limit from the left does not equal the limit from the right, the limit as x approaches 0 of f(x) does not exist (DNE).