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

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

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

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

- \infty

Solution steps

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

x \to a lim [c \cdot f (x)] = c \cdot x \to a lim f (x)

= 2 \cdot 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}} )}

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

Cancel the common factor:

x

= 2 \cdot 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

= 2 (- \infty)

Apply Infinity Property:

c \cdot (- \infty) = - \infty

= - \infty

x \to \infty lim ([0, + x])

x \to 1 + lim (x^{\frac{3}{x - 1}})

x \to 0 - lim (x (\frac{1}{x}))

x \to 4 lim (3 x (x + 5))

x \to \frac{π}{2} lim (cot^{2} (x))

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