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

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

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

3) What is the function value as  and is it the same as the result from statement one and two?

We can observe that  , as  approaches  from the left and from the right.

Since an absolute value function is involved, first rewrite the expression as a piecewise function.

 is a left-sided limit, so in we are interested in what happens as the function approached 0 from values x<0.

The function f(x) is  for , so the limit has value . 

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

Therefore, we can observe that   as  approaches  from the left.

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

Therefore, we can observe that   as  approaches  from the left.

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   as  approaches  from the right.

To evaluate the limit, simply substitute what we are approaching (5) into the function:

That we are approaching 5 from the left has no impact on the final answer. 

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

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

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

3) What is the function value as  and is it the same as the result from statement one and two?

Therefore, we can observe that   as  approaches  from the left and 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   as  approaches  from the right.

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

Thus, we can observe that   as   approaches  from the left.

Because this is a one sided limit, we can't simply subsitute the value into the function being evaluated. The function is piecewise, changing from one function  to a different one at . We are evaluating the limit from the right side, meaning greater than 2. We actually reach a value at 2, which is  (approaching 2 from the left - in other words, at values barely greater than 2). 

Thus,