Evaluate the following limits. \lim_{x\to\infty}(2+\frac{4}{x})^{2x}

Burhan Hopper

Burhan Hopper

Answered question

2021-10-29

Evaluate the following limits.
limx(2+4x)2x

Answer & Explanation

comentezq

comentezq

Skilled2021-10-30Added 106 answers

Given:
limx(2+4x)2x
Calculation:
limx(2+4x)2x if we substitute limit x in this we get indeterminate form 2
Let's define y=(2+4x)2x
And take the log on both side and we will do some simplification
lny=ln(2+4x)2x
And take the log on both side and we will do some simplification
lny=ln(2+4x)2x
lny=2xln(2+4x)
lny=ln(2+4x)12x
Taking limit x on both side
limx[lny]=ln(2+4x)12x
limx[lny]=ln(2+4)12
limx[lny]=ln(2+0)0=ln(2)0=
We know that
t=elny
This means we can do original limit as follow
limx(2+4x)2x=limx[y]=limxelny=elimx[lny]e=
Therefore limx(2+4x)2x=
Jeffrey Jordon

Jeffrey Jordon

Expert2022-08-30Added 2605 answers

Answer is given below (on video)

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