some concrete limits

Дьявол

1. The problem statement, all variables and given/known data

Hello! I got few questions, about limits.
º [tex]\lim_{x \rightarrow 0}(\frac{sin(x)}{x})=1[/tex]
If I take values for x close to zero I get:
f(x)=sinx/x
f(0.1)=0.017453283
f(0.01)=0.017453292
as I can see it is not even close to 1.
What is the problem? Where I am doing wrong?

º [tex]\lim_{x \rightarrow \infty}(\frac{1}{x})=0[/tex]
Now, for all integers I agree that [tex]\lim_{x \rightarrow \infty}(\frac{1}{x})=0[/tex] (thanks to HallsofIvy for [itex]\infty[/itex]), but what for 1/2, 1/3, 1/4 ?
[tex]\lim_{x \rightarrow \infty}(\frac{1}{x})=\frac{1}{1/4}=4[/tex] and not 0 ?

º [tex]\lim_{x \rightarrow 0^-}(e^{\frac{1}{x}})[/tex]
How will I find the bound of the expression above?

2. Relevant equations


3. The attempt at a solution
[tex]\lim_{x \rightarrow 0^-}(e^{\frac{1}{x}})[/tex]
I understand that x<0 (so the values for x are tending to 0 from the left side), and
[tex]\lim_{n \rightarrow \infty}(x_n)=0[/tex]

For example, I know how to find the bound for:
[tex]\lim_{x \rightarrow 2^+}(\frac{x}{x-2})[/tex]
D=R\{2}
x_n>2
[tex]\lim_{n \rightarrow \infty}(x_n)=2[/tex]
[tex]x_n-2>0[/tex]
[tex]\lim_{n \rightarrow \infty}(x_n-2)=0[/tex]
so that:
[tex]\lim_{x \rightarrow 2^+}(\frac{x}{x-2})=\lim_{n \rightarrow \infty}(\frac{x_n}{x_n-2})=\frac{2}{\lim_{n \rightarrow \infty}(2-x_n)}=+\infty[/tex]

Thanks in advance.

Mark44

Are you using a calculator to do these? If so, I think your calculator is in degree mode. It needs to be in radian mode. As the values of x get smaller, the value of your expression will get closer to 1.
The limit is for x growing very large, so you shouldn't concern yourself with small values of x. On the other hand,
[tex]\lim_{x \rightarrow 0^+}(\frac{1}{x})=\infty[/tex]
which is more related to what you're doing with 1/2, 1/3, and so on.
As x approaches 0 from the negative side, 1/x approaches neg. infinity, so e^(1/x) approaches 0. Do you need more explanation than that?

Дьявол

Thanks for the post Mark44.

Yes, I was using calculator in degree mode. Now with radian mode everything is all right.

For the second one. Sorry, I wasn't so clear. I was learning about the number "e". So for one task (example):
[tex]\lim_{x \rightarrow \infty}(1+\frac{1}{x-3})=\lim_{x \rightarrow \infty}(1+\frac{1}{x-3})^{(x-3)+3}=\lim_{t \rightarrow \infty}(1+\frac{1}{t})^t*\lim_{t \rightarrow \infty}(1+\frac{1}{t})^3=e*(1+0)^3=e[/tex]
As we can see they put [tex]\lim_{t \rightarrow \infty}(\frac{1}{t})=0[/tex]. How is this possible? What about for t=1/2,1/3,1/4 ? It wouldn't be zero in that case.

Mark44, sorry for misunderstanding again. Yes I understand all of that, but how will I "show" or "prove" that. Aren't there any calculations?
Thanks in advance.

Mark44

For your first limit above, I think you are missing an exponent of x. In other words, I think it should be:
[tex]\lim_{x \rightarrow \infty}(1+\frac{1}{x-3})^x[/tex]
Now I think I understand what you're saying. The first limit was as x approached infinity and involved an expression with (x - 3). They substituted t = x - 3 and changed the limit variable from x to t (as x gets very large, so does t).

In the third limit expression (before they took the limit), there are two factors:
[tex]}(1+\frac{1}{t})^t[/tex] and
[tex]}(1+\frac{1}{t})^3[/tex].
The second one is straightforward to evaluate in the limit, and turns out to be just 1. If you multiply it out before taking the limit, you have 1 + 3*1/t + 3*1/t^2 + 1/t^3, which approaches 1 as t gets large.

The first one is more tricky, and you can't just say that 1/t approaches 0 as t gets large. There are two competing effects going on: the base, 1 + 1/t, is getting closer to 1, but the exponent t is getting larger. The net effect is that (1 + 1/t)^t approaches the number e as t gets large. If I recall correctly, one of the definitions of e is precisely this limit.

Дьявол

Now I understand. I misjudged [itex]\infty[/itex].

But for [tex]\lim_{x \rightarrow 0^-}(e^{1/x})[/tex], I need to "show", "explain" the result of the limit. How will I do that?

Regards.

HallsofIvy

For x very close to 0, but negative, say -0.00001, 1/x is -100000, a very large negative number. What is e^-100000? what is e ^{any very large negative number}?

Дьявол

Thanks for the post. It definitely will tend to zero.

And what about [tex]\lim_{x \rightarrow 0^+}(\frac{1}{1+e^{1/x}})[/tex]?

[tex]\lim_{x \rightarrow 0^+}(e^{1/x})=\infty[/tex] ?

Mark44

For the first one, as x approaches 0 (from the right), 1/x grows without bound (approaches infinity), so 1 + 1/x also grows without bound, which makes the fraction approach 0.

For the second, that's the right value.