$\text { Evaluate the limit } \lim _ { x \rightarrow 0 ^ { + } } \frac { \ln \left( x ^ { 2 } \right) } { x ^ { 7 } } \text { using L'Hopital's Rule if necessary. }$
A) $- \infty$
B) $\infty$
C) $\frac { 2 } { 7 }$
D) 0
E) $\frac { 7 } { 2 }$

Question

$\text { Evaluate the limit } \lim _ { x \rightarrow 0 ^ { + } } \frac { \ln \left( x ^ { 2 } \right) } { x ^ { 7 } } \text { using L'Hopital's Rule if necessary. }$ 
A) $- \infty$
B) $\infty$
C) $\frac { 2 } { 7 }$
D) 0
E) $\frac { 7 } { 2 }$

Quizplus · Accepted Answer

Applying L'Hopital's Rule, differentiate the numerator and denominator separately. The derivative of $\ln(x^2)$ is $\frac{2}{x}$, and the derivative of $x^7$ is $7x^6$. Thus, the limit becomes $\lim _ { x \rightarrow 0 ^ { + } } \frac { 2 } { 7x^7 }$, which approaches $-\infty$ as $x$ approaches $0$ from the positive side.

A) −∞- \infty−∞ B) ∞\infty∞ C) 27\frac { 2 } { 7 }72​

A) $- \infty$ B) $\infty$ C) $\frac { 2 } { 7 }$