Determine the following limits.
lim x→−∞ e^x ...

Question

Determine the following limits.

lim x→−∞ e^x sin x

Accepted Answer

Welcome back, everyone. Evaluate the limit. Limit as X approaches negative infinity of 11 e to the power of X multiplied by cosine of X. We're given 4 answer choices A says 0, B-1, C1 and D11/2. So what we're going to do is rewrite the limit. Limits X approaches negative infinity of 11 E to the power of X cosine x. And as usually, well, we essentially begin by direct substitution. If we directly substitute negative infinity for every X that we can see, well, we're going to get 11 multiplied by each the power of negative infinity multiplied by cosine of negative infinity. When we think about this expression, well eats the power of negative infinity. Essentially approaches 0, right? This is an exponential function with a negatively. Or basically infinitely small exponent. Specifically, it is negative, right and it's approaching infinity, and this essentially means that the exponential term is approaching zero. Now, cosine. Oscillates between -1 and 1. This is the range of cosine. When we approach negative infinity, cosine simply keeps oscillating. So to clearly illustrate that the limit is zero, we're going to apply the squeeze theorem. First of all, let's write the range of cosine X. Cosine X is between -1 and 1 inclusive. What we're going to do from here is multiply. Our inequality by 11 e to the power of X. And this gives us -11 E to the power of X is less than or equal to 11 to the power of X cosine X, which is less than or equal to. 11 E to the power of X and now we can apply the limit. To our inequality, this means that The limit as x approaches negative infinity. Of -11 eats the power of x. Must be less than or equal to. The limit as x approaches negative infinity of our function, which is 11 e to the power of x cosine x, which is less than or equal to. The limit as x approaches negative infinity of 11 eats the power of X. Let's evaluate the first limit. Now if x approaches negative infinity, E X approaches 0, right? So we're going to say that the first limit is going to be equal to 0. We have a -11 multiplied by 0, which gives us 0. And now for the final limit in our inequality. Once again 11 multiplied by e X as X approaches negative infinity, this limit is equal to 0 because we have an exponential function and X approaches an infinitely small number, right, or specifically. Negative infinity because we're also looking at a negative number. So now we have our limit squeezed between two zeros and according to the squeeze theorem, this means that the limit between the two zeros is also going to be equal to 0. Therefore, the final answer to this problem corresponds to the answer choice A 0. Thank you for watching.

Determine the following limits.
lim x→−∞ e^x sin x

Key Concepts

Limits at Infinity

Exponential Functions

Trigonometric Functions

Watch next