L'Hopital's rule for solving limit

[mousesgr]

1. lim [(1+x)^(1/x) - e ] / x
x ->0

2. lim [sin(2/x)+cos(1/x)]^x
x -> inf

help...

 

[HallsofIvy]

Homework should be posted in the homework forum- and you should show us what you have tried to do. Have you considered L'Hopital's rule?

 

[mousesgr]

qs 1 is not 0/0 or inf/inf from
how do consider L'Hopital's rule?

 

[HallsofIvy]

Do you KNOW "L'Hopital's rule"? It is specifically for limit problems where you get things like these.

If you have $$lim_{x->a} \frac{f(x)}{g{x}}$$ where f(a)= 0 and g(a)= 0, then the limit is the same as $$lim_{x->a}\frac{\frac{df}{dx}}{\frac{dg}{dx}}$$.

If you get things like $$0^0$$ or $$\infty^{\infty}$$ (as your second limit), you can take logarithms to reduct to the first case.

 

[VietDao29]

qs 1 is not 0/0 or inf/inf from
how do consider L'Hopital's rule?

#1 is in form 0 / 0.
Since:
$$\lim_{x \rightarrow 0} (1 + x) ^ {\frac{1}{x}} = e$$
So the numerator will tend to 0 as x approaches 0. The denominator also tends to 0. So it's 0 / 0.
You can use L'Hopital's rule to solve for #1.
-------------------
#2 is $$1 ^ \infty$$
First, you can try to take logs of both sides.
So let $$y = \lim_{x \rightarrow \infty} \left[ \sin \left( \frac{2}{x} \right) + \cos \left( \frac{1}{x} \right) \right] ^ x$$. So:
$$\ln y = \ln \left\{ \lim_{x \rightarrow \infty} \left[ \sin \left( \frac{2}{x} \right) + \cos \left( \frac{1}{x} \right) \right] ^ x \right\} = \lim_{x \rightarrow \infty} \ln \left[ \sin \left( \frac{2}{x} \right) + \cos \left( \frac{1}{x} \right) \right] ^ x$$
$$= \lim_{x \rightarrow \infty} x \ln \left[ \sin \left( \frac{2}{x} \right) + \cos \left( \frac{1}{x} \right) \right] = \lim_{x \rightarrow \infty} \frac{\ln \left[ \sin \left( \frac{2}{x} \right) + \cos \left( \frac{1}{x} \right) \right]}{\frac{1}{x}}$$
This is 0 / 0. So again, you can apply L'Hopital's rule to find the limit.
Viet Dao,