In this explainer, we will learn how to evaluate limits of trigonometric functions.

Limits are a useful tool for helping us understand the shape of a function around a value; it is one of the fundamental building blocks of calculus. We can find the limit of any trigonometric function by using direct substitution.

### Definition: Evaluating the Limit of Trigonometric Functions

If is in the domain of a trigonometric function, then we can evaluate its limit at by direct substitution. In particular, for any ,

- ,
- .

For any in the domain of ,

- .

These results allow us to evaluate the limit of many trigonometric expressions. However, there are examples that we cannot evaluate. For example, consider , where is measured in radians. If we attempt to evaluate this limit using direct substitution, we find an indeterminate form, which means we need to evaluate this limit in a different manner. One way of doing this is to sketch the graph of .

From the sketch, we can see that as the values of approach 0 from either side, the outputs of the function approach 1. Hence, the sketch indicates that . We can also see this by constructing a table.

0 | 0.001 | 0.01 | 0.1 | ||||||

0.99833 | 0.99998 | 0.99999 | 0.99999 | 0.99998 | 0.99833 |

Once again, the table suggests that as the values of approach 0 from either side, the outputs of the function approach 1. It is worth noting that we can show a similar result when is measured in degrees; however, when taking limits, we almost always use radians. So, unless otherwise stated, we will assume that the limit of any trigonometric functions involves angles measured in radians. This gives us the following result.

### Theorem: Limit of a Trigonometric Expression

If is measured in radians, then

We can use this result to show an even more general result. Let . We substitute into the limit result . Note that as , both and . This gives us

Taking the factor of out of this limit and rearranging gives us

It is worth noting this result also holds when . We can summarize this as follows.

### Theorem: Limit of a Trigonometric Expression

If is measured in radians and , then

Let’s see an example of using this result to evaluate the limit of a trigonometric expression.

### Example 1: Finding Limits Involving Trigonometric Functions

Evaluate .

### Answer

Since this limit involves the quotient of trigonometric functions, we can attempt to evaluate this limit by direct substitution

This gives us an indeterminate form, which means that we cannot evaluate this limit by direct substitution. Instead, we will use the fact that if is measured in radians and is a real constant, then . Although the question does not state that is measured in radians, when taking limits, we almost always work in radians, so we will assume this for the question. We can rewrite the limit as follows:

Assuming that both limits exist, we can write this as the product of two limits:

We take the reciprocal of the second limit by using the power rule for limits to get provided the limit exists and is nonzero. We can then evaluate both of these limits by using our limit result,

The first limit has and the second has . Hence,

There are two more useful limit results involving trigonometric functions that we can find by investigating their graph or by using a table. Consider the following sketches of and , where is measured in radians.

In the first diagram, we see that as the values of approach 0, the outputs approach 1. So, the sketch suggests . Similarly, in the second diagram, as the values of approach 0, we can see that the outputs approach 0. So, the sketch suggests that . This gives us the following results.

### Theorem: Limit of a Trigonometric Expression

If is measured in radians, then

- ,
- .

As with the limit result involving sine, we can use substitution to find a limit result where the argument is a constant multiple. If , using we have

Taking out the constant factor of and rearranging gives

Similarly, if , using we have

Taking out the constant factor of and rearranging gives

We can summarize this as follows.

### Theorem: Limit of a Trigonometric Expression

If is measured in radians and , then

- ,
- .

Let’s see an example of how we can apply these limit results to evaluate the limit of a trigonometric expression.

### Example 2: Finding Limits Involving Trigonometric Functions

Determine .

### Answer

Since this limit involves a trigonometric function, we can attempt to evaluate this limit by direct substitution:

This gives us an indeterminate form, which means we cannot evaluate this limit by direct substitution. Instead, we will use the fact that if is measured in radians and , then

To apply this result, we simplify our limit as follows:

Hence, .

In our next example, we will use a limit result involving the tangent and sine functions to evaluate the limit of a trigonometric function.

### Example 3: Finding Limits Involving Trigonometric Functions

Find .

### Answer

Since this is the limit of a trigonometric and algebraic expression, we can attempt to evaluate this limit by direct substitution:

Since this is an indeterminate form, we cannot determine the value of this limit from direct substitution. Instead, we will rewrite this limit in terms of limits we can evaluate. Namely, for any real constant and measured in radians,

We can rewrite the limit in the question as follows:

We can evaluate each of these limits separately. First, we recall that if is measured in radians and constant , . Using this result, we have

Next, we recall that if is measured in radians and , then

Hence,

Substituting the values of these limits into the equation gives us

Therefore, .

In our next example, we will combine a trigonometric identity with the limit results of trigonometric functions to evaluate a limit.

### Example 4: Finding Limits Involving Trigonometric Functions

Find .

### Answer

Since this is the limit of a trigonometric and algebraic expression, we can attempt to evaluate this limit by direct substitution:

Since this is an indeterminate form, we cannot determine the value of this limit from direct substitution. Instead, we will rewrite this limit in terms of limits we can evaluate. We rewrite the limit as follows:

To evaluate this limit, we will use the substitution . As approaches , will approaches 0. This gives us

Recall that . We can use this to rewrite the limit as

Finally, we recall that .

Hence,

Therefore, .

In our final example, we will use these limit results to evaluate the limit of a reciprocal trigonometric expression.

### Example 5: Finding Limits Involving Trigonometric Functions

Find .

### Answer

Since this is the limit of a trigonometric and algebraic expression, we can attempt to evaluate this limit by direct substitution. However, 0 is not in the domain of this function. Instead, we will rewrite the limit by first using the reciprocal trigonometric identities:

We can then rewrite this in terms of the limit results. If is measured in radians and ,

So, we have

Applying the limit results, we conclude that

Hence, .

Let’s finish by recapping some of the important points from this explainer.

### Key Points

- We can evaluate the limit of any trigonometric function at by direct substitution if a is in its domain.
- If is measured in radians, we have the following trigonometric limit results:
- ,
- ,
- .

- If is measured in radians and , we have the following trigonometric limit results:
- ,
- ,
- .

- We can use these results to evaluate the limit of trigonometric functions.