What is the limit as x approaches-3-of x/(x^2-9) ?

The limit as x approaches-3-of x/(x^2-9) is -infinity

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limit as x approaches-3-of x/(x^2-9)

x \to - 3 - lim (\frac{x}{x ^{2} - 9})

- \infty

Solution steps

x \to - 3 - lim (\frac{x}{x ^{2} - 9})

Apply the following algebraic property

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

\frac{x}{x ^{2} - 9} = \frac{x}{x ^{2} ( 1 - \frac{9}{x ^{2}} )}

= x \to - 3 - lim (\frac{x}{x ^{2} ( 1 - \frac{9}{x ^{2}} )})

Cancel the common factor:

x

= x \to - 3 - lim (\frac{1}{x ( - \frac{9}{x ^{2}} + 1 )})

For

x

approaching

- 3

from the left

, x < - 3 \Rightarrow x (- \frac{9}{x ^{2}} + 1) < 0

The denominator is a negative quantity approaching 0 from the left

= - \infty

x \to 1 lim (3 x^{2} + 2)

x \to 3 lim (\frac{7}{( x - 3 ) ^{3}})

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

x \to \infty lim (b \cdot e^{- x})

x \to 1 lim (3 x^{2} - x)

What is the limit as x approaches-3-of x/(x^2-9) ?
The limit as x approaches-3-of x/(x^2-9) is -infinity