Find the limit?

Question

#lim_"x➞0" (2x + e^(10x))^(5/x)#

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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#