What is the limit as x approaches-1-of x/((x+1)^3) ?

The limit as x approaches-1-of x/((x+1)^3) is infinity

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limit as x approaches-1-of x/((x+1)^3)

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

\infty

Solution steps

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

Apply the following algebraic property

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

\frac{x}{( x + 1 ) ^{3}} = \frac{x}{( x ( 1 + \frac{1}{x} ) ) ^{3}}

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

Simplify

\frac{x}{( x ( 1 + \frac{1}{x} ) ) ^{3}} : \frac{1}{x ^{2} ( \frac{1}{x} + 1 ) ^{3}}

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

For

x

approaching

- 1

from the left

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

The denominator is a positive quantity approaching 0 from the right

= \infty

\int 1 + 4 r^{2} d r

\frac{d}{d x} (1 + 7 x)

\int \frac{( 1 - z )}{z z} d z

\int - 2 x^{2} + 2 d x

7 sin^{2} (t)

What is the limit as x approaches-1-of x/((x+1)^3) ?
The limit as x approaches-1-of x/((x+1)^3) is infinity