Find the limit?

Question

$lim_"x➞0" (2x + e^(10x))^(5/x)$

1052 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-07-28T12:16:46

$e^60$

Given limit:

$\lim_{x\to 0}(2x+e^{10x})^{5/x}$

$=\exp \lim_{x\to 0}ln(2x+e^{10x})^{5/x}$

$=\exp \lim_{x\to 0}(5/x)ln(2x+e^{10x})$

$=\exp5 \lim_{x\to 0}\frac{ln(2x+e^{10x})}{x}$

Using L'Hospital's rule for $0/0$ form

$=\exp5 \lim_{x\to 0}\frac{d/dxln(2x+e^{10x})}{d/dx(x)}$

$=\exp5 \lim_{x\to 0}\frac{\frac{1}{2x+e^{10x}}d/dx(2x+e^{10x})}{1}$

$=\exp5 \lim_{x\to 0}\frac{2+10e^{10x}}{2x+e^{10x}}$

$=\exp5\cdot \frac{2+10e^{0}}{2\cdot 0+e^{0}}$

$=exp(5\cdot 12)$

$=e^60$