I found a set of problems of limits which i can't seem to work my way around.
I tried using the natural log and then applying L'Hospital's rule but I can't seem to make it work.
The problem was to find the limit of the function of
$$\left(\frac{\tan x}{x}\right)^\frac1x$$
As x approaches 0
Please help me as there are some more problems like this. I cannot think of ways to evaluate them
Best Answer
L'Hospital's rule is not the alpha and omega of limits computations!
To determine the limit of the logarithm, use Taylor's formula at order $3$ for the tangent: $$\frac{\tan x}x=\frac{x+\cfrac{x^3}3+o(x^3)}x=1+\frac{x^2}3+o(x^2),$$ so that $$\frac1x\log\Bigl(\frac{\tan x}x\Bigr)=\frac1x\log\Bigl(1+\frac{x^2}3+o(x^2)\Bigr)=\frac1x\Bigl(\frac{x^2}3+o(x^2)\Bigr)=\frac{x}3+o(x)\to 0$$ and finally $\;\biggl(\dfrac{\tan x}x\biggr)^{\!\tfrac 1x}$ tends to $1$ as $x$ tends to $0$.