What are the Limits of These Functions at Zero and Infinity?

[Deadleg]

Homework Statement

1. Find the limit of $$\lim_{x\rightarrow 0} \frac{1}{xe^{\frac{1}{x}}}$$
2. " " " " $$\lim_{x\rightarrow\infty} \frac {x}{\log_e x}$$

Homework Equations

$$\lim_{x\rightarrow\infty} \frac{N}{x} = 0$$

$$\lim_{x\rightarrow n} x+a = \lim_{x\rightarrow n} x + \lim_{x\rightarrow n} a$$ etc

The Attempt at a Solution

1. I put in values of x close to 0, and as I approached from above I got values very close to 0, but when I approached from below the numbers became massively large and negative ($$f(-0.1)=-220264, f(-0.01)=-2.688\times10^{45}$$). The answer in my book is zero, but my numbers say there is no limit as values of x approaching 0 do not approach the same number. Have I missed something out or is the book wrong?

2. In the book the answer is "no limit", but I can't think of a way to evaluate it to prove it. The only thing I've thought of is dividing by x, but that did nothing and ended up going in circles :/

 

[Defennder]

Have you learned L Hospital's rule yet? Use it for both. Note that you have to express 1. in the correct form before you can use it.

 

[Deadleg]

No, we don't learn that this year. This is last year high school stuff, and I was sick when the class was taught it so I'm trying to get through it myself. The notes up to this exercise in the book simply goes over what a limit is, evaluating by algebraic manipulation or solve for values of x and draw a graph/table, the equations listed above and cases of being careful with moduli.

 

[Defennder]

Ok then perhaps we can take this somewhat intuitively. Consider the first question. As x->0, analyse the term in the denominator xe^(1/x). x will approach 0 and e^(1/x) will approach infinity, right? So we have two limits going in the opposite directions (very roughly speaking). But which one of these 2 would "reach its limit" faster? Which term would dominate?

Alternatively, think of 1. as $$\frac{1/x}{e^{1/x}}$$. Draw a graph of the numerator and that of the denominator on the same sketch. Which one would dominate as x->0?

The second one you can also think of it intuitively. Look at the graph of y=x and y=ln x. What happens when x->infinity? Which one diverges faster?

 

[Deadleg]

Ah I get it now, thanks!