Question

Given the function f(x) =
`(|x|)/(x)` , find
`\lim_(x \to \00^-) f(x) and \lim_(x \to \00^+) f(x) and \lim_(x \to \00) f(x)`

Answer 1

`\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

Explanation:

Step 1: Define

`\displaystyle f(x) = (|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).