What is the limit as x approaches-1-of x/(x^2-1) ?

The limit as x approaches-1-of x/(x^2-1) is -infinity

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limit as x approaches-1-of x/(x^2-1)

x \to - 1 - lim (\frac{x}{x ^{2} - 1})

- \infty

Solution steps

x \to - 1 - lim (\frac{x}{x ^{2} - 1})

Apply the following algebraic property

: a + b = a (1 + \frac{b}{a})

\frac{x}{x ^{2} - 1} = \frac{x}{x ^{2} ( 1 - \frac{1}{x ^{2}} )}

= x \to - 1 - lim (\frac{x}{x ^{2} ( 1 - \frac{1}{x ^{2}} )})

Cancel the common factor:

x

= x \to - 1 - lim (\frac{1}{x ( - \frac{1}{x ^{2}} + 1 )})

For

x

approaching

- 1

from the left

, x < - 1 \Rightarrow x (- \frac{1}{x ^{2}} + 1) < 0

The denominator is a negative quantity approaching 0 from the left

= - \infty

What is the limit as x approaches-1-of x/(x^2-1) ?
The limit as x approaches-1-of x/(x^2-1) is -infinity