Question #0a300

Question

1533 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-09-20T20:50:14

$lim_(x to 0^-) sqrt (3 + arctan(1/x)) = sqrt (3 - pi/2)$

$lim_(x to 0^+) sqrt (3 + arctan(1/x)) = sqrt (3 + pi/2)$

$lim_(x to 0) sqrt (3 + arctan(1/x))$

now we can plot this to see where this limit is going....

Desmos

....so we can see that

$lim_(x to 0) sqrt (3 + arctan(1/x)) \ \ \$ is two sided

We know that:

$lim_(x to 0^-) arctan(1/x) = - pi/2$

$lim_(x to 0^) arctan(1/x) = pi/2$

So
$lim_(x to 0^-) sqrt (3 + arctan(1/x)) = sqrt (3 - pi/2)$

$lim_(x to 0^+) sqrt (3 + arctan(1/x)) = sqrt (3 + pi/2)$