# Limits at infinity of trigometric function

• Willian93
In summary, the limit at infinity of a trigonometric function is the value that the function approaches as the input (x) gets infinitely large. To determine this limit, one can use the rules of limits and the properties of trigonometric functions. It is possible for the limit at infinity to be undefined if the function oscillates or does not approach a specific value. The limit at infinity differs from the limit as x approaches infinity in that the former focuses on the function's overall behavior while the latter focuses on its behavior as x gets larger. Finally, trigonometric functions can have different limits at positive and negative infinity if their behavior differs in the positive and negative x-directions.
Willian93

## Homework Statement

lim (x→pi)〖sin(x-pi)/(x-pi)〗

## Homework Equations

i don't know if we should use trig identity

sin(a-b)= sinA cos B- CosA sin B

## The Attempt at a Solution

i use identities to solve that, i did not get the answer. i tried to multiply by conjugate, did not work also.

Last edited:

You could use a substitution, u = x - $\pi$, and that limit should be familiar.

## 1. What is the limit at infinity of a trigonometric function?

The limit at infinity of a trigonometric function is the value that the function approaches as the input (x) gets infinitely large. This can also be referred to as the behavior of the function as x approaches infinity.

## 2. How do you determine the limit at infinity of a trigonometric function?

To determine the limit at infinity of a trigonometric function, you can use the rules of limits, such as the fact that the limit of a sum is equal to the sum of the limits. You can also use the properties of trigonometric functions, such as the fact that sine and cosine are bounded between -1 and 1.

## 3. Can the limit at infinity of a trigonometric function be undefined?

Yes, the limit at infinity of a trigonometric function can be undefined if the function oscillates or does not approach a specific value as x gets infinitely large. In this case, we say that the limit does not exist.

## 4. What is the difference between the limit at infinity and the limit as x approaches infinity?

The limit at infinity is the value that the function approaches as x gets infinitely large, while the limit as x approaches infinity is the value that the function approaches as x gets closer and closer to infinity. The former focuses on the behavior of the function as a whole, while the latter focuses on the behavior of the function as x gets larger and larger.

## 5. Can trigonometric functions have different limits at positive and negative infinity?

Yes, trigonometric functions can have different limits at positive and negative infinity if the function has different behavior in the positive and negative x-directions. For example, the limit at positive infinity for the function f(x) = sin(x) is 1, while the limit at negative infinity is -1.

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