Question #0a300

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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) #