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Exercises Limits and derivatives

Example 9 from Precalculus, 12.2 Finding Limits: Properties of Limits

Evaluating the Limit of a Quotient with Absolute Values

Evaluate $\lim_{x \to 7} \frac{| x - 7 |}{x - 7} .$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
The function is undefined at $x = 7,$ so we will try values close to 7 from the left and the right.

Left-hand limit: $\frac{| 6.9 - 7 |}{6.9 - 7} = \frac{| 6.99 - 7 |}{6.99 - 7} = \frac{| 6.999 - 7 |}{6.999 - 7} = - 1$
Right-hand limit: $\frac{| 7.1 - 7 |}{7.1 - 7} = \frac{| 7.01 - 7 |}{7.01 - 7} = \frac{| 7.001 - 7 |}{7.001 - 7} = 1$

Since the left- and right-hand limits are not equal, there is no limit.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Evaluate $\lim_{x \to 6^{+}} \frac{6 - x}{| x - 6 |} .$

How did it go?

PreviousEvaluating the Limit of a Quotient of a Function by Factoring Next Identifying Discontinuities

From Precalculus (1st edition) by OpenStax (Rice University), section 12.2, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.