# Calculus 2 : Limit Concepts

## Example Questions

### Example Question #1 : Limit Concepts

Evaluate the limit:

Does Not Exist

Explanation:

Directly evaluating the limit will produce an indeterminant answer of .

Rewriting the limit in terms of sine and cosine, , we can try to manipulate the function in order to utilize the property .

Multiplying the function by the arguments of the sine functions, , we can see that the limit will be .

### Example Question #2 : Limit Concepts

Find the limit of  as  approaches infinity.

Inconclusive

Explanation:

The expression  can be rewritten as .

Recall the Squeeze theorem can be used to solve for the limit.  The sine function has a range from , which means that the range must be inside this boundary.

Multiply the  term through.

Take the limit as  approaches infinity for all terms.

Since the left and right ends of this interval are zero, it can be concluded that  must also approach to zero.

### Example Question #3 : Limit Concepts

Determine the limit.

Explanation:

To determine, , graph the function  and notice the direction from the left and right of the curve as it approaches .

Both the left and right direction goes to negative infinity.

### Example Question #4 : Limit Concepts

Which of the following is true?

If  exists, then  and  both exist.

If  and , then  exists.

and  exist if and only if  exists.

If neither  nor  exist, then  also doesn't exist.

If  and , then  exists.

Explanation:

If  and , then  exists.

This can be proven rigorously using the  definition of a limit, but it is most likely beyond the scope of your class.

### Example Question #5 : Limit Concepts

Determine the limit:

Explanation:

Isolate the constant in the limit.

The limit property .

Therefore:

### Example Question #6 : Limit Concepts

Evaluate the limit, if possible:

Explanation:

To evaluate , notice that the inside term  will approach infinity after substitution.  The inverse tangent of a very large number approaches to .

### Example Question #7 : Limit Concepts

Evaluate the following limit:

Explanation:

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

which becomes

Evaluating the limit, we approach .

### Example Question #8 : Limit Concepts

Evaluate the following limit:

Explanation:

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

which becomes

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

### Example Question #9 : Limit Concepts

Evaluate the following limit:

Explanation:

To evaluate the limit, first pull out the highest power term out of the numerator and denominator (so essentially you are pulling 1):

As you can see, the  and  terms as they approach infinity go to zero. What is left over is .

### Example Question #10 : Limit Concepts

Evaluate the following limit: