Determine the following limits or explain why they do not exist?

f(x)= sinx/x

a) lim (as x goes to infinity) f(x)

b) lim (as x goes to pi from the left) ln(f(x))

I got the answers, but I'd really like the simplest explanation possible as I am trying to figure out "how else" the professor can ask this on the test. Also, for part b, his explanation was that approaching zero with a very small, finite # is DNE. how does ln(sinx/x) equal that?

Thanks for any help!

Re: Determine the following limits or explain why they do not exist?

$\displaystyle \lim_{x \to \infty} \frac{\sin{x}}{x} = 0$

as $\displaystyle x \to \infty$, $\displaystyle \sin{x}$ oscillates between -1 and 1 while the x in the denominator gets very large ... hence, you essentially have a relatively fixed value in the numerator and a very large value in the denominator ... the value of the fraction approaches 0.

$\displaystyle \lim_{x \to \pi^-} \ln\left(\frac{\sin{x}}{x}\right) = -\infty $ , so the limit DNE.

as $\displaystyle x \to \pi^-$ , $\displaystyle \sin{x} \to 0^+$ and $\displaystyle x \to \pi$ ... therefore $\displaystyle \frac{\sin{x}}{x} \to 0^+$

now ... that happens to the basic log function as its argument approaches 0 from the right?

Re: Determine the following limits or explain why they do not exist?

Thanks! I'm still sort of confused. As x approaches pi from left, would ln(sinx/x) be equivalent to ln(0/small) in order to be DNE or something else?

If question b had said as x approaches pi from the right, would it still be negative infinity? I'm thinking it would be positive infinity since it would get to negative infinity from the left, and the right get infinitely bigger.

Re: Determine the following limits or explain why they do not exist?

Quote:

Originally Posted by

**Steelers72** f(x)= sinx/x

a) lim (as x goes to infinity) f(x)

b) lim (as x goes to pi from the left) ln(f(x))

If $\displaystyle f(x)=\frac{g(x)}{x}$ and $\displaystyle g$ is a bounded function then $\displaystyle \lim _{x \to \infty } f(x) = 0$.

If $\displaystyle \lim _{x \to 0^ - } \ln(|x|) = - \infty $

Re: Determine the following limits or explain why they do not exist?

Quote:

As x approaches pi from left, would ln(sinx/x) be equivalent to ln(0/small)

???

as $\displaystyle x \to \pi^-$ ... $\displaystyle \ln\left(\frac{0}{\pi} \right) \implies \ln(0) \to -\infty$ , the limit DNE

approaching $\displaystyle \pi$ from the right would not be possible ... the argument for the log function would approach 0 from the left, which cannot happen since the domain of the log function is greater than 0.

Re: Determine the following limits or explain why they do not exist?

Ohhh ok.

So, basically, nonzero(or bounded)/infinitely large = 0

and

Infinitely large/nonzero = DNE

Am I on the right track?