Infinite limits and finding a limit

[Violagirl]

I did not know how to start this problem so if anyone could show me that would be great!

1. Find the limit (if it exists).

lim [1/(x+1)]-1
_________________
x
x-->0


And for these is it possible to figure out these problems below without a calculator?

Find the one sided limit.

2. lim csc 2x
_______
x
x-->0+



3. lim ln(sin x)
x--->0+

 

[galactus]

1. Find the limit (if it exists).

lim [1/(x+1)]-1
_________________
x
x-->0

Is this what you mean

\(\displaystyle \lim_{x\to 0}\frac{\frac{1}{x+1}-1}{x}\)

If so, do some algebra and realize it is equal to \(\displaystyle \lim_{x\to 0}\frac{-1}{x+1}\)

What does that equal when x=0?.

 

[Violagirl]

Yep that is what I meant. And I could not figure how to alternate it to the next step. :?

 

[galactus]

3. lim ln(sin x)
x--->0+

Take note that as \(\displaystyle x\to 0\), then \(\displaystyle x\sim sin(x)\)

That is why \(\displaystyle \lim_{x\to 0}\frac{sin(x)}{x}=1\)

Since we can say \(\displaystyle \lim_{x\to 0^{+}}ln(sin(x))\sim\lim_{x\to 0^{+}}ln(x)\)

Now, going with this, we can proceed as follows:

Let \(\displaystyle u=\frac{1}{x}, \;\ u\to\infty \;\ as \;\ x\to 0^{+}\)

\(\displaystyle \lim_{x\to 0^{+}}ln(x)=\lim_{u\to\infty}\left(ln\frac{1}{u}\right)=\lim_{u\to\infty}(-ln(u))=-\lim_{u\to \infty}ln(u)=-\infty\)

 

[mmm4444bot]

… I could not figure how to [simplify] it …

Algebra is a prerequisite for calculus. Are you in school or self-studying? :?

 

[galactus]

2. lim csc 2x
_______
x
x-->0+


Treat this one kind of like the third problem.

We can rewrite as \(\displaystyle \lim_{x\to 0^{+}}\frac{1}{xsin(2x))}\)

But \(\displaystyle sin(x)\sim x \;\ as \;\ x\to 0\)

So, we have:

\(\displaystyle \lim_{x\to 0}\frac{1}{x^{2}}=\infty\)

 

[jillwile]

how to you find the limit to infinity of xsin(1/x)
I graphed it and checked the table and saw it goes to 1 but how to I explain it algebraically?

 

[Deleted member 4993]

how to you find the limit to infinity of xsin(1/x)
I graphed it and checked the table and saw it goes to 1 but how to I explain it algebraically?

substitute u = 1/x

Then it becomes a famous identity.

 

[BigGlenntheHeavy]

Use a rotating magnetic field to focus a narrow beam of gravitons. These in turn fold space-time, consistent with the Weyl tensor dynamic, until space-time curvature becomes infinite and you create a singularity.galactus

galactus, in quantum theory, every force must have a carrier. The photon, for example, is the carrier of the electromagnetic force. The gulon is the carrier of the force that holds hadrons together. The graviton is the "projected carrier" of the gravitational force. Now as far as I know gravity waves are of yet to be detected or am I out of date on this one.

 

[galactus]

It is a line from a sci-fi film I liked called "Event Horizon". I just thought it sounded 'scientific and futuristic". Yes, as far as I know, gravitons are theoretical. I may be wrong, perhaps they have been isolated. They sure use them a lot in sci-fi films though.