Determine the following limits at infinity.

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Question

Determine the following limits at infinity.

lim x→−∞ x^−11

Accepted Answer

Below there, today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Find the limit of the given function at infinity. The limit as X approaches negative infinity of X to the power of -51. Awesome. So it appears for this particular problem, we're trying to figure out the limit of the given function at infinity. Awesome. So with that in mind, let us read off our multiple choice answers to see what our final answer might be. A is negative infinity, B is 0, C is 1, and D is positive infinity. Awesome. So first off, in order to find the limit of the function, which we'll call F of X, and we'll say that F of X is equal to x to the power of negative 51, so in order to find the limit of this function as X approaches, and let's make a note of this, as X approaches negative infinity, we need to follow 5 simple steps. So in 5 simple steps we can easily solve this problem. So let's start with our first step. So for our first step, we need to recall and note that the function F of X, we need to note that F of X is equal to X of the power of -51, is a power function where the exponent is a negative odd integer. And for our second step, we need to note that for any power function where X is X to the power of N, where N is a negative integer, the limit as X approaches, so we need to note that as X approaches negative infinity. So, OK, let's like, let's take a moment here to pause. So, for any power function where X to the power of N, where N in this case is a negative integer, the limit as X approaches negative infinity will be equal to 0. And this is because as X becomes very, very large number. X to the power of N will become a very, very small number due to the negative exponent making it a fraction. So now we can write our limit expression for our third step and and as we should recall, we should be able to write our limit expression as the limit as x approaches negative infinity of X to the power of negative 51. And we now for our fourth step, we need to evaluate the limit, and we need to note that since X to the power of -51 becomes a fraction. With a very large denominator, as X approaches negative infinity, the value of the function will approach 0. So therefore, our last step we can say in conclusion we should be able to recall and write that the limit as X approaches negative infinity of X to the power of negative 51 is equal to 0, and that is our final answer. Hooray, we did it, and this corresponds to multiple choice answer letter B. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Determine the following limits at infinity.

lim x→−∞ x^−11

Key Concepts

Limits at Infinity

Polynomial Functions

Negative Exponents

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