The limit we are given is one sided, meaning we are approaching our x value from one side; in this case, the negative sign exponent indicates that we are approaching 3 from the left side, or using values slightly less than three on approach.

This corresponds to the part of the piecewise function for values less than 3. When we substitute our x value being approached, we get

$\displaystyle (3)^2+1=10$

The limit $\displaystyle \lim_{x\to 0^+}$ indicates that we are trying to find the value of the limit as $\displaystyle x$ approaches to zero from the right side of the graph.  

From right to left approaching $\displaystyle x=0$, the limit approaches to 1 even though the value at $\displaystyle x=0$ of the piecewise function does not exist.

The answer is $\displaystyle 1$.

Examining the graph of the function above, we need to look at three things:

1) What is the limit of the function as it approaches zero from the left?

2) What is the limit of the function as it approaches zero from the right?

3) What is the function value at zero and is it equal to the first two statements?

If we look at the graph we see that as $\displaystyle x$ approaches zero from the left the $\displaystyle y$ values approach zero as well. This is also true if we look the values as $\displaystyle x$ approaches zero from the right. Lastly we look at the function value at zero which in this case is also zero.

Therefore, we can observe that $\displaystyle \lim_{x\rightarrow 0}f(x)=0$ as $\displaystyle x$ approaches $\displaystyle 0$.

Examining the graph above, we need to look at three things:

1) What is the limit of the function as $\displaystyle x$ approaches zero from the left?

2) What is the limit of the function as $\displaystyle x$ approaches zero from the right?

3) What is the function value as $\displaystyle x=0$ and is it the same as the result from statement one and two?

Therefore, we can determine that $\displaystyle \lim_{x\rightarrow 0}f(x)$ does not exist, since $\displaystyle x$ approaches two different limits from either side : $\displaystyle -\infty$ from the left and $\displaystyle \infty$ from the right. 

Examining the graph, we want to find where the graph tends to as it approaches zero from the right hand side. We can see that there appears to be a vertical asymptote at zero. As the x values approach zero from the right the function values of the graph tend towards positive infinity.

Therefore, we can observe that $\displaystyle \lim_{x\rightarrow 0^{+}}f(x)=\infty$  as $\displaystyle x$ approaches $\displaystyle 0$ from the right.

Examining the graph, we can observe that does not exist, as   is not continuous at . We can see this by checking the three conditions for which a function is continuous at a point :

1. A value exists in the domain of

2. The limit of exists as approaches

3. The limit of at is equal to

Given , we can see that condition #1 is not satisfied because the graph has a vertical asymptote instead of only one value for and is therefore an infinite discontinuity at .

We can also see that condition #2 is not satisfied because approaches two different limits:  from the left and from the right.

Based on the above, condition #3 is also not satisfied because is not equal to the multiple values of .

Thus, does not exist.

$\displaystyle \begin{align*}&\text{Although actually observing a function at an infinite value}\\&\text{of x is impossible, we can often determine via observation if}\\&\text{the function approaches a certain limit. Looking at the values}\\&\text{the function approaches as the absolute value of x increases}\\&\text{gives a clue. Things to look out for are if the function continuously}\\&\text{climbs/decreases, flattens, or has unchecked oscillatory behavior.}\\&\text{If the function continuously climbs/decreases, it likely has}\\&\text{a limit that approaches positive/negative infinity. If it flattens,}\\&\text{the limit is a real number. If it oscillates without control,}\\&\text{the limit does not exist.}\\&\text{The limit of }f(-\infty)\text{ is most likely }\text{Nonexistent.}\end{align*}$

$\displaystyle \begin{align*}&\text{Although actually observing a function at an infinite value}\\&\text{of x is impossible, we can often determine via observation if}\\&\text{the function approaches a certain limit. Looking at the values}\\&\text{the function approaches as the absolute value of x increases}\\&\text{gives a clue. Things to look out for are if the function continuously}\\&\text{climbs/decreases, flattens, or has unchecked oscillatory behavior.}\\&\text{If the function continuously climbs/decreases, it likely has}\\&\text{a limit that approaches positive/negative infinity. If it flattens,}\\&\text{the limit is a real number. If it oscillates without control,}\\&\text{the limit does not exist.}\\&\text{The limit of }f(-\infty)\text{ is most likely }0\end{align*}$

$\displaystyle \begin{align*}&\text{Although actually observing a function at an infinite value}\\&\text{of x is impossible, we can often determine via observation if}\\&\text{the function approaches a certain limit. Looking at the values}\\&\text{the function approaches as the absolute value of x increases}\\&\text{gives a clue. Things to look out for are if the function continuously}\\&\text{climbs/decreases, flattens, or has unchecked oscillatory behavior.}\\&\text{If the function continuously climbs/decreases, it likely has}\\&\text{a limit that approaches positive/negative infinity. If it flattens,}\\&\text{the limit is a real number. If it oscillates without control,}\\&\text{the limit does not exist.}\\&\text{The limit of }f(\infty)\text{ is most likely }-5\end{align*}$

$\displaystyle \begin{align*}&\text{Although actually observing a function at an infinite value}\\&\text{of x is impossible, we can often determine via observation if}\\&\text{the function approaches a certain limit. Looking at the values}\\&\text{the function approaches as the absolute value of x increases}\\&\text{gives a clue. Things to look out for are if the function continuously}\\&\text{climbs/decreases, flattens, or has unchecked oscillatory behavior.}\\&\text{If the function continuously climbs/decreases, it likely has}\\&\text{a limit that approaches positive/negative infinity. If it flattens,}\\&\text{the limit is a real number. If it oscillates without control,}\\&\text{the limit does not exist.}\\&\text{The limit of }f(-\infty)\text{ is most likely }\text{Nonexistent.}\end{align*}$