Expert answers to "Find the following limits or state that they..."

BenoguigoliB

Answered question

2021-11-05

Find the following limits or state that they do not exist.

lim_{x \to 1^{+}} \frac{x - 1}{\sqrt{x^{2} - 1}}

Obiajulu

Skilled2021-11-06Added 98 answers

Given:

lim_{x \to 1^{+}} \frac{x - 1}{\sqrt{x^{2} - 1}}

for evaluating given limit we substitute x=1+h, where

h \to 0

so,

lim_{x \to 1^{+}} \frac{x - 1}{\sqrt{x^{2} - 1}} = lim_{h \to 0^{+}} \frac{1 + h - 1}{\sqrt{{(1 + h)}^{2} - 1}}

= lim_{h \to 0^{+}} \frac{h}{\sqrt{1^{2} + 2 (1) (h) + h^{2} - 1}}

= lim_{h \to 0^{+}} \frac{h}{\sqrt{1 + 2 h + h^{2} - 1}}

= lim_{h \to 0^{+}} \frac{h}{\sqrt{h^{2} + 2 h}}

= lim_{h \to 0^{+}} \frac{h}{h \sqrt{1 + \frac{2}{h}}}

= lim_{h \to 0^{+}} \frac{1}{\sqrt{1 + \frac{2}{h}}}

= \frac{1}{\sqrt{1 + \infty}}

= 0

hence, given limit is equal to 0.

Do you have a similar question?

Recalculate according to your conditions!

Didn't find what you were looking for?