Question

Evaluate using a substitution:
lim x->2+ of arctan((1/(x-2))

Answer 1

\[\lim_{x \rightarrow 2+} \arctan (\frac{ 1 }{ x-2 })\]

Answer 2

so what happens if you substitute x with 2?

Answer 3

The number inside the parenthesis becomes undefined (1/0)

Answer 4

and what is the arctangent of undefined?

Answer 5

okay... here's a hint:
\[\tan = \frac{\sin}{\cos}\]
that will become undefined if cosine is 0... so what value of x will make cos x = 0?

Answer 6

pi/2 makes cos x =0

Answer 7

right. \[\tan (\frac \pi 2) = \frac{\sin (\frac \pi 2)}{\cos ( \frac \pi 2)} \implies \tan (\frac \pi 2) = \frac 10 \implies \text{undefined}\]
do you get it now?

Answer 8

I understand that but how does this help if its arctan?

Answer 9

if the tangent of pi/2 is undefined...then what do you think is the arctangent of undefined?

Answer 10

OHHH its pi/2

Answer 11

indeed

Answer 12

But like is that how i wud show work? Like on an exam instead of reasoning with teh tangent is tehre a way to show taht its indeed pi/2?

Answer 13

actually..you don't ahve to show the solution of the tangent...arctangent of (1/0) is already automatic

Answer 14

i just showed you the solution of the tangent for you to see

Answer 15

B/C im supposed to evaluate using subsition. So what shud I write on paper as my work then?

Answer 16

Wait is like arctan of undefined always pi/2 b/c thats just smthg we need to memorize just like sin(90) = 1

Answer 17

THANKS NEVER MIND I GET IT :D

Answer 18

arctan (undefined) is not automatic 90 degrees by the way... it's 90 degrees in this case ebcause the numerator is 1 and 1 can only be got with sin (90)