Find the value of the limit $\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 3 } - 1 } .$
A)0
B)2
C)
$\frac { 1 } { 3 }$
D)8
E) $\frac { 1 } { 4 }$
F) $\frac { 1 } { 8 }$
G)
$\frac { 1 } { 32 }$
H) $\frac { 1 } { 2 }$

Question

Find the value of the limit $\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 3 } - 1 } .$
A)0
B)2
C)
$\frac { 1 } { 3 }$
D)8
E) $\frac { 1 } { 4 }$ 
F) $\frac { 1 } { 8 }$
G)
$\frac { 1 } { 32 }$ 
H) $\frac { 1 } { 2 }$

Quizplus · Accepted Answer

The limit can be simplified by factoring the denominator as $x^3 - 1 = (x - 1)(x^2 + x + 1)$, which gives $\lim _ { x \rightarrow 1 } \frac { x - 1 } { (x - 1)(x^2 + x + 1) }$. Canceling $x - 1$ from the numerator and denominator, we get $\lim _ { x \rightarrow 1 } \frac { 1 } { x^2 + x + 1 }$. Substituting $x = 1$, we find the limit to be $\frac { 1 } { 3 }$.

Find the Value of the Limit lim⁡x→1x−1x3−1.\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 3 } - 1 } .limx→1​x3−1x−1​.

Find the Value of the Limit $\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 3 } - 1 } .$