"Find the following limit: limx→0xexe3x−1" was answered by our experts

ankarskogC

Answered question

2021-03-04

Find the following limit:

lim_{x \to 0} \frac{x e^{x}}{e^{3 x} - 1}

casincal

Skilled2021-03-05Added 82 answers

Given:

lim_{x \to 0} \frac{x e^{x}}{e^{3 x} - 1}

Now Applying limits:

lim_{x \to 0} \frac{x e^{x}}{e^{3 x} - 1} = \frac{0 (e^{0})}{e^{3 (0)} - 1} = \frac{0}{0}

Hence it gives indeterminate form

\frac{0}{0}

Now by L-Hospital Rule:

lim_{x \to 0} [\frac{\frac{d}{d x} (x e^{x})}{\frac{d}{d x} (e^{3 x} - 1)}] = lim_{x \to 0} [\frac{x e^{x} + e^{x}}{3 e^{3 x}}]

= [\frac{0 (e^{0}) + e^{0}}{3 e^{3 (0)}}]

[= \frac{1}{3}]

Hence,

lim_{x \to 0} \frac{x e^{x}}{e^{3 x} - 1} = \frac{1}{3}

Jeffrey Jordon

Expert2022-04-01Added 2605 answers

Answer is given below (on video)

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