Zion Wheeler

2022-06-25

Evaluate limit of $\underset{x\to 0}{lim}\frac{\left(1+x{\right)}^{\left(1/5\right)}-\left(1-x{\right)}^{\left(1/5\right)}}{x}$

pyphekam

Expert

Start by adding and subtracting 1 in the numerator:
$\underset{x\to 0}{lim}\frac{\left(1+x{\right)}^{\frac{1}{5}}-1+1-\left(1-x{\right)}^{\frac{1}{5}}}{x}$
Now split up the limit like this:
$\underset{x\to 0}{lim}\frac{\left(1+x{\right)}^{\frac{1}{5}}-1}{x}+\underset{x\to 0}{lim}\frac{\left(1-x{\right)}^{\frac{1}{5}}-1}{-x}$
Using the substitution $u=-x$ on the second limit, you should see that both limits are the definition of the derivative of $f\left(x\right)={x}^{\frac{1}{5}}$ at $x=1$
${f}^{\prime }\left({x}_{0}\right)=\underset{h\to 0}{lim}\frac{f\left({x}_{0}+h\right)-f\left({x}_{0}\right)}{h}$
${f}^{\prime }\left(1\right)=\underset{h\to 0}{lim}\frac{\left(1+h{\right)}^{\frac{1}{5}}-{1}^{\frac{1}{5}}}{h}$
Using the power rule, we know that ${f}^{\prime }\left(x\right)=\frac{1}{5}{x}^{-\frac{4}{5}}.$ So, the answer is $2{f}^{\prime }\left(1\right)=2\left(\frac{1}{5}\left(1{\right)}^{-\frac{4}{5}}\right)=\overline{)\frac{2}{5}.}$
Alternatively, you can see this by using the alternative definition ${f}^{\prime }\left(1\right)=\underset{x\to 0}{lim}\frac{f\left(1+x\right)-f\left(1-x\right)}{2x}$ and multiplying the numerator and denominator of our original limit by 2

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