Evaluated at $\displaystyle x = 1$, the numerator and the denominator are both equal to 0, as shown below:

$\displaystyle 8^{x}- 8 = 8^{1}- 8 = 8 - 8 = 0$

$\displaystyle 2^{x}- 2 = 2^{1}- 2 = 2 - 2 = 0$

So a straightforward substitution will not work. L'Hospital's rule will work here, but an easier way is to note that

$\displaystyle 8 = 2 ^{3}$ and $\displaystyle 8^{x} = \left ( 2 ^{3} \right )^{x} = 2 ^{3x} = \left ( 2 ^{x} \right )^{3}$.

So the expression can be rewritten - and solved - as follows:

$\displaystyle \lim_{x\rightarrow 1} \frac{8^{x}- 8}{2^{x}- 2}$

$\displaystyle = \lim_{x\rightarrow 1} \frac{\left (2^{x} \right )^{3}- 2^{3}}{2^{x}- 2}$

$\displaystyle = \lim_{x\rightarrow 1} \frac{\left (2^{x}- 2 \right) \left [ \left ( 2^{x} \right ) ^{2} +2 \cdot 2^{x} + 2 ^{2} \right ] }{2^{x}- 2}$

$\displaystyle = \lim_{x\rightarrow 1} \left [ \left ( 2^{x} \right ) ^{2} +2 \cdot 2^{x} + 2 ^{2} \right ]$

$\displaystyle = \left [ \left ( 2^{1} \right ) ^{2} +2 \cdot 2^{1} + 2 ^{2} \right ] = 2^{2} + 2 \cdot 2 + 2^{2} = 4 + 4 + 4 = 12$

Directly evaluating the limit will produce an indeterminant answer of $\displaystyle \frac{0}{0}$.

Rewriting the limit in terms of sine and cosine, $\displaystyle \lim_{y \rightarrow 0} \frac{\sin 2y}{\cos 2y} \cdot \frac{1}{\sin 5y}$, we can try to manipulate the function in order to utilize the property $\displaystyle \lim_{x \rightarrow 0} \frac{\sin x}{x}=1$.

Multiplying the function by the arguments of the sine functions, $\displaystyle \lim_{y \rightarrow 0} \frac{\sin 2y}{2y} \cdot \frac{1}{\cos 2y} \cdot \frac{5y}{\sin 5y} \cdot \frac{2y}{5y}$, we can see that the limit will be $\displaystyle (1)(1)(1)\left(\frac{2}{5}\right) = \left(\frac{2}{5} \right )$.

$\displaystyle \lim_{x\rightarrow \frac{\pi}{2}^{-} } \cos x = \cos \frac{\pi}{2} = 0$

and 

$\displaystyle \lim_{x\rightarrow \frac{\pi}{2}^{-} } \tan x = \infty$,

so we cannot solve this by substituting.

However, we can rewrite the expression:

$\displaystyle \lim_{x\rightarrow \frac{\pi}{2}^{-} } \cos x \cdot \frac{\sin x}{\cos x}$

$\displaystyle = \lim_{x\rightarrow \frac{\pi}{2}^{-} } \sin x = \sin \frac{\pi}{2} = 1$

The expression $\displaystyle \frac{sin^5(x)}{x^5}$ can be rewritten as $\displaystyle \frac{1}{x^5}sin^5(x)$.

Recall the Squeeze theorem can be used to solve for the limit.  The sine function has a range from $\displaystyle [0,1]$, which means that the range must be inside this boundary.

$\displaystyle 0\leq sin^5(x) \leq 1$

Multiply the $\displaystyle \frac{1}{x^5}$ term through.

$\displaystyle 0\left(\frac{1}{x^5}\right)\leq \left(\frac{1}{x^5}\right)sin^5(x) \leq 1\left(\frac{1}{x^5}\right)$

Take the limit as $\displaystyle x$ approaches infinity for all terms.

$\displaystyle \lim_{x \to \infty}0\leq \lim_{x \to \infty}\frac{sin^5(x)}{x^5} \leq \lim_{x \to \infty}\frac{1}{x^5}$

$\displaystyle 0\leq \lim_{x \to \infty}\frac{sin^5(x)}{x^5} \leq 0$

Since the left and right ends of this interval are zero, it can be concluded that $\displaystyle \lim_{x \to \infty}\frac{sin^5(x)}{x^5}$ must also approach to zero.

The correct answer is 0.

To determine, $\displaystyle \lim_{x \to 0}-\frac{1}{x^2}$, graph the function $\displaystyle y=-\frac{1}{x^2}$ and notice the direction from the left and right of the curve as it approaches $\displaystyle x=0$.

Both the left and right direction goes to negative infinity.

The answer is:  $\displaystyle -\infty$

If $\displaystyle \small \lim _{x\to a} f(x)$ and $\displaystyle \small \lim _{x\to a} g(x)$, then $\displaystyle \small \lim _{x\to a} f(x)+g(x)$ exists.

This can be proven rigorously using the $\displaystyle \small \epsilon-\delta$ definition of a limit, but it is most likely beyond the scope of your class. 

Isolate the constant in the limit.

$\displaystyle \lim_{x \to \infty}\frac{3x^n}{9n!} = \frac{1}{3}\lim_{x \to \infty}\frac{x^n}{n!}$

The limit property $\displaystyle \lim_{x \to \infty}\frac{x^n}{n!}=0$.

Therefore:

$\displaystyle \frac{1}{3}\lim_{x \to \infty}\frac{x^n}{n!} = \frac{1}{3}(0)=0$

To evaluate $\displaystyle \lim_{x\to \infty}tan^{-1}(x^2)$, notice that the inside term $\displaystyle x^2$ will approach infinity after substitution.  The inverse tangent of a very large number approaches to $\displaystyle \frac{\pi}{2} \textup{ radians}$.

The answer is $\displaystyle \frac{\pi}{2}$.

The first step is to factor out the highest degree term from the polynomial on top and bottom (essentially pulling out 1):

$\displaystyle \lim_{x\rightarrow \infty }\frac{x^2}{x^2}\left(\frac{1+\frac{1}{x^2}}{3}\right)$

which becomes

$\displaystyle \lim_{x\rightarrow \infty }\left(\frac{1+\frac{1}{x^2}}{3}\right)$

Evaluating the limit, we approach $\displaystyle \frac{1}{3}$.

To evaluate the limit, first pull out the largest power term from top and bottom (so we are removing 1, in essence):

$\displaystyle \lim_{x\rightarrow \infty }\frac{x^2(\frac{1}{x})}{x^2(2+\frac{3}{x}+\frac{1}{x^2})}$

which becomes

$\displaystyle \lim_{x\rightarrow \infty }\frac{(\frac{1}{x})}{(2+\frac{3}{x}+\frac{1}{x^2})}$

Plugging in infinity, we find that the numerator approaches zero, which makes the entire limit approach 0.