# Limit with e

1. Jan 14, 2009

### Dell

how do i solve this limit

Lim (ex+sinx)1/sinx
x->0

its almost oiler (1+x)1/x, at 1st looking at it i thought it was simple e, because ex when x->0 is 1, but i see i need to somehow turn the ex into 1,,, any ideas??

2. Jan 14, 2009

### NoMoreExams

I would take ln of both sides and then use L'Hopital's

3. Jan 14, 2009

### Dell

both of which sides?? can you show/describe what you mean

4. Jan 14, 2009

### tim_lou

take the ln and use L'Hopital.

If you just want to evaluate it, expand it first order in x should be good.

Be careful though, for small x,
$$e^x\approx 1+x$$
and
$$\sin(x) \approx x$$
so your "intuitive" conclusion (1+x)^{1/x} doesn't hold.

5. Jan 14, 2009

### Dell

what do i do with the ln,?? take ln on what??

6. Jan 14, 2009

### NoMoreExams

You have

$$y = \lim_{x \rightarrow 0} \left(e^{x} + sin(x)\right)^{\frac{1}{sin(x)}}$$

If you take ln of both sides of the expression you have

$$ln(y) = \lim_{x \rightarrow 0} \frac{ln(e^{x} + sin(x))}{sin(x)}$$

Now do L'Hopital's and go from there, remember in the end you want to solve for y.