Write a proof for each limit using the $\varepsilon-\delta$ definition of a limit. $\lim _{x \rightarrow 3}(4 x)=12$

Find the limit L. Then use the $\varepsilon-\delta$ definition to prove that the limit is L. $\lim _{x \rightarrow 9} \sqrt{x}$

Find the limit L. Then use the $\varepsilon-\delta$ definition to prove that the limit is L. $\lim _{x \rightarrow 1}(x+4)$

Find the limit, if it exists, or show that the limit does not exist. lim (x,y,z) to (0,0,0) xy+yz/x^2+y^2+z^2

Find the limit $L$. Then use the $\varepsilon-\delta$ definitio to prove that the limit is $L$.

$$\lim\limits_{x\to 6} |x-6|$$

Express the integral as a limit of Riemann sums. Do not evaluate the limit.

$$\int_0^{2 \pi} x^2 \sin x d x$$

Find the limit of the sequence or state that the limit does not exist.

$$t_n=\cos \left(\frac{n \pi}{2}\right)$$

Find the limit, if it exists. If the limit does not exist, explain why. lim x tends to -6 2x+12/|x+6|

Find the limit, if it exists, or show that the limit does not exist. lim (x,y) tends to (1,0) xy-y/(x-1)^2+y^2

Find the limit, if it exists, or show that the limit does not exist. $\lim _{(x, y) \rightarrow(0,0)} \frac{x y}{\sqrt{x^{2}+y^{2}}}$

Find the limit, if it exists, or show that the limit does not exist. lim (x,y) tends to (0,0) xy^4/x^2+y^8

Determine the limit of the sequence or show that the sequence diverges. If it converges, find its limit. $a_{n}=\frac{(n !)^{2}}{(2 n) !}$

Use L’Hopital’s rule to evaluate the given limit if the limit is an indeterminate form.

$$\lim _{x \rightarrow \infty} \sqrt{x} e^{-x}$$

Find the limit of the following sequences or determine that the limit does not exist. $\left\{ \frac { n ^ { 3 } } { n ^ { 4 } + 1 } \right\}$