How do you evaluate the limit (sin(5x))/x as x approaches 0?

Ivan Waters

Ivan Waters

Answered question

2023-03-24

How to evaluate the limit sin ( 5 x ) x as x approaches 0?

Answer & Explanation

Tomas Cobb

Tomas Cobb

Beginner2023-03-25Added 7 answers

Given lim x 0 sin ( 5 x ) x
We want to use the previously mentioned limit, so we need to have θ = 5 x
To get 5x in the denominator, we'll multiply by 5 5
lim x 0 sin ( 5 x ) x = lim x 0 5 sin ( 5 x ) 5 x
Now, outside the limit, factor the 5 in the numerator.
= 5 lim x 0 sin ( 5 x ) 5 x
As x 0 , 5 x 0 so we have:
= 5 ( 1 ) = 5
(The limit is 5.)
Hence, using lim θ 0 sin θ θ = 1 and some other tools.

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