Question

how to solve this indeterminate form by l'hospital rule??

Answer 1

\[\lim_{x \rightarrow 0} \frac{ \tan^{-1} x }{ \log (1+x^2) }\]

Answer 2

at 0 the function is evaluated to be 0/0, so yes we can use l'hopital's rule
the derivative of atan is 1/(1+x^2)
the derivative of log(1+x^2):
chain rule
1/(1+x^2) * 2x

Answer 3

@absurdism but that 1/(1+x^2) divided by 1/(1+x^2) * 2x = 1/2x
lim x ->0 1/2x = infinity ... isnt it ??

Answer 4

exactly, so there's no 2-sided limit that exists.

Answer 5

and the one-sided limits are positive and negative infinity

Answer 6

@absurdism could you show me how to right the answer ?? as the answer given in the book is 1

Answer 7

the limit does not exist.
if the book says the answer is 1, then either you've got the question wrong or the book is wrong, unless i'm making some silly mistake

Answer 8

even i got the same thing as you got, so i wanted to check if i was making any mistake :(

Answer 9

right, i think the book is wrong.

Answer 10

because wolframalpha agrees the limit does not exist