Find $\lim _ { x \rightarrow 0 ^ { + } } x ^ { 3 x }$
A) $e ^ { 3 }$
B) 0
C) 1
D) e

Question

Quizplus · Accepted Answer

To find the limit of $x^{3x}$ as $x$ approaches 0 from the positive side, we can rewrite the expression using the property that $a^{b} = e^{b\ln(a)}$, which gives us $e^{3x\ln(x)}$. The limit can then be found by evaluating the limit of the exponent $3x\ln(x)$ as $x$ approaches 0 from the positive side. Using L'Hôpital's Rule to evaluate this limit, we differentiate the numerator and the denominator with respect to $x$, which gives us $\frac{3(1+\ln(x))}{1/x}$, and as $x$ approaches 0, this expression approaches 0. Therefore, the limit of the exponent $3x\ln(x)$ is 0, and thus $e^{3x\ln(x)}$ approaches $e^0$, which is 1.

Find lim⁡x→0+x3x\lim _ { x \rightarrow 0 ^ { + } } x ^ { 3 x }limx→0+​x3x

Find $\lim _ { x \rightarrow 0 ^ { + } } x ^ { 3 x }$