Find each of the following limits. If the limit is not finite, indicate or for one- or two-sided limits, as appropriate. lim_{xrightarrowinfty}frac{4x^3-2x-1}{x^2-1}

How to solve $\underset{n\to \mathrm{\infty }}{lim}{\left(\frac{a^{\frac{1}{n}}+b^{\frac{1}{n}}}{2}\right)}^n$

Limit ${\displaystyle \underset{x\to \frac{\pi }{3}}{lim}\frac{\sin (x-\frac{\pi }{3})}{1-2\cos \left\{x\right\}}}$

Evaluate the following limits: $\underset{y\to 0}{lim}\frac{1-\cos (7y)}{2y}$

Given that ${\displaystyle \underset{x\to a}{lim}f\left(x\right)=0,\underset{x\to a}{lim}g\left(x\right)=0,\underset{x\to a}{lim}h\left(x\right)=1,\underset{x\to a}{lim}p\left(x\right)=\mathrm{\infty },\underset{x\to a}{lim}q\left(x\right)=\mathrm{\infty }}$ which of the following limits are indeterminate forms? For those that are not an indeterminate form, evaluate the limit where possible. ${\displaystyle \underset{x\to a}{lim}p\frac{x}{q}\left(x\right)}$.

Find the following limits or state that they do not exist.
${\displaystyle \underset{x\to 4}{lim}\frac{x^2-16}{4-x}}$

I would like to evaluate the limit
${\displaystyle \underset{h\to 0}{lim}\left(\frac{1-\cos \left\{5h\right\}}{\cos \left\{7h\right\}-1}\right)}$
I'm having a hard time doing this, however. When you try evaluating directly, of course, it comes out in indeterminate form. I tried using the identity
${\displaystyle \underset{h\to 0}{lim}\left(\frac{1-\cos h}{h}\right)=0}$
by breaking the limit up into two limits, by dividing and multiplying by h, but that only takes me to ${\displaystyle 0\cdot 10}$. What can I do to evaluate this?

Analyze the continuity of the function below. Determine the type(s) of discontinuity and justify your answer.
$f(x)=\{\begin{array}{lll}x,& if& x<0\\ 3,& if& x=0\\ x^2,& if& 0<x<3\\ x+6,& if& x\ge 3\end{array}$