Evaluate limit of lim x → 0 ( 1 + x ) ( 1 / 5 ) − ( 1 − x ) ( 1 / 5 ) x

Name: Evaluate limit of lim x → 0 ( 1 + x ) ( 1 / 5 ) − ( 1 − x ) ( 1 / 5 ) x
Uploaded: 2022-02-23T17:22:57+00:00
Description: Evaluate limit of lim x → 0 ( 1 + x ) ( 1 / 5 ) − ( 1 − x ) ( 1 / 5 ) x

Question

Evaluate limit of        lim          x      →      0                  (      1      +      x              )                  (          1                      /                    5          )                    −      (      1      −      x              )                  (          1                      /                    5          )                      x

pyphekam · Accepted Answer

Start by adding and subtracting 1 in the numerator:

lim_{x \to 0} \frac{(1 + x)^{\frac{1}{5}} - 1 + 1 - (1 - x)^{\frac{1}{5}}}{x}

Now split up the limit like this:

lim_{x \to 0} \frac{(1 + x)^{\frac{1}{5}} - 1}{x} + lim_{x \to 0} \frac{(1 - x)^{\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 (x) = x^{\frac{1}{5}}

at

x = 1

f^{'} (x_{0}) = lim_{h \to 0} \frac{f (x_{0} + h) - f (x_{0})}{h}

f^{'} (1) = lim_{h \to 0} \frac{(1 + h)^{\frac{1}{5}} - 1^{\frac{1}{5}}}{h}

Using the power rule, we know that

f^{'} (x) = \frac{1}{5} x^{- \frac{4}{5}} .

So, the answer is

2 f^{'} (1) = 2 (\frac{1}{5} (1)^{- \frac{4}{5}}) = \frac{2}{5} .

Alternatively, you can see this by using the alternative definition

f^{'} (1) = lim_{x \to 0} \frac{f (1 + x) - f (1 - x)}{2 x}

and multiplying the numerator and denominator of our original limit by 2