Limits Involving Trigonometric Functions
The trigonometric functions sine and cosine have four important limit properties:
![](https://s3.amazonaws.com/dev-hmhco-vmg-craftcms-public/_cliffsnotes/assets/39116.gif)
You can use these properties to evaluate many limit problems involving the six basic trigonometric functions.
Example 1: Evaluate
.
Substituting 0 for x, you find that cos x approaches 1 and sin x − 3 approaches −3; hence,
![](https://s3.amazonaws.com/dev-hmhco-vmg-craftcms-public/_cliffsnotes/assets/39118.gif)
Example 2: Evaluate
Because cot x = cos x/sin x, you find
The numerator approaches 1 and the denominator approaches 0 through positive values because we are approaching 0 in the first quadrant; hence, the function increases without bound and
and the function has a vertical asymptote at x = 0.
Example 3: Evaluate
Multiplying the numerator and the denominator by 4 produces
![](https://s3.amazonaws.com/dev-hmhco-vmg-craftcms-public/_cliffsnotes/assets/39120.gif)
Example 4: Evaluate
.
Because sec x = 1/cos x, you find that
![](https://s3.amazonaws.com/dev-hmhco-vmg-craftcms-public/_cliffsnotes/assets/39122.gif)