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

Zion Wheeler

Zion Wheeler

Answered

2022-06-25

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

Answer & Explanation

pyphekam

pyphekam

Expert

2022-06-26Added 27 answers

Start by adding and subtracting 1 in the numerator:
lim x 0 ( 1 + x ) 1 5 1 + 1 ( 1 x ) 1 5 x
Now split up the limit like this:
lim x 0 ( 1 + x ) 1 5 1 x + lim x 0 ( 1 x ) 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 1 5 at x = 1
f ( x 0 ) = lim h 0 f ( x 0 + h ) f ( x 0 ) h
f ( 1 ) = lim h 0 ( 1 + h ) 1 5 1 1 5 h
Using the power rule, we know that f ( x ) = 1 5 x 4 5 . So, the answer is 2 f ( 1 ) = 2 ( 1 5 ( 1 ) 4 5 ) = 2 5 .
Alternatively, you can see this by using the alternative definition f ( 1 ) = lim x 0 f ( 1 + x ) f ( 1 x ) 2 x and multiplying the numerator and denominator of our original limit by 2

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