Determine the following limits.
lim x→−∞ (2x^-...

Question

Determine the following limits.

lim x→−∞ (2x^-8+ 4x³)

Accepted Answer

Welcome back, everyone. We want to find the limit as X approaches negative infinity of 18 X to the negative 12 plus 5 X to the 9th. AS 23, B 5, C infinity, and the D negative infinity. Now here we know that we can rewrite our limit for our separate terms in the expression. In other words, this is going to be equal to the limit, OK, as X approaches negative infinity of 18 x 12, OK, plus the limit as X approaches negative infinity of 5 X 9th. Now we can also break out over constants here and we can rewrite this then as 18 multiplied by the limit as x approaches negative infinity of 1 divided by X to the 12th. Plus 5 multiplied by the limit as X approaches negative infinity of X 9th. Now what is going to help us solve for both of these limits? Well, recall, OK, that by the theorem for limits of infinity powers, let me just make that note down here. Recall that for a positive integer N. OK. As it relates to our first expression, our first term. The limit as x approaches negative infinity of 1 divided by x N equals 0. Therefore, we know this term, this first term is going to be equal to 18, multiplied by 0, which equals 0. OK? So we've been able to solve for the first one. What about our second term? Where we also know that for that positive integer n, the limit as x approaches negative infinity of X to the power of n is going to be equal to negative infinity when n is odd. Now for this expression, n would be equal to 99 is odd. Therefore, the theorem for limits at infinity powers applies. In other words, this is going to be equal to 5 multiplied by negative infinity, which is still negative infinity, and 0 plus negative infinity will be equal to negative infinity. Thus, this is going to be the limit of our expression. Therefore, D is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Determine the following limits.
lim x→−∞ (2x^-8+ 4x³)

Key Concepts

Limits at Infinity

Dominant Terms

Polynomial Growth Rates

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