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

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Answer 1 · 2016-11-07T21:45:12

Use $lim_(thetararr0)sintheta/theta = 1$ and some other tools.

$lim_(xrarr0)sin(5x)/x$

We'd like to use the limit mentioned above, so we need to have $theta = 5x$

To get $5x$ in the denominator, we'll multiply by $5/5$

$lim_(xrarr0)sin(5x)/x = lim_(xrarr0)(5sin(5x))/(5x)$

Now factor the $5$ in the numerator outside the limit.

$= 5lim_(xrarr0)sin(5x)/(5x)$

As $xrarr0$ , $5xrarr0$ so we have:

$= 5(1) = 5$

(The limit is $5$ .)

How do you evaluate the limit sin(5x)/xsin(5x)/x as x approaches 00?