Determine

Question

Determine $\lim_{x\rightarrow\infty}f\left(x\right)$ and $\lim_{x\rightarrow-\infty}f\left(x\right)$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).

$f (x) = \frac{40 x^{5} + x^{2}}{16 x^{4} - 2 x}$

Accepted Answer

Welcome back, everyone solve the following limits and identify the horizontal asymptotes. If any for the function F of X is equal to nine X fourth plus two, X divided by three X cubed minus X squared, we want to evaluate two limits limit of F of X. When X approaches positive and negative infinity were also given for answer choices. ABC and D, they give us the answers to each limit and the expression of the horizontal Asymptote. So let's begin this problem by solving for the first limit, we want to evaluate the limit of F of X when X approaches positive infinity. And our function F of X is nine multiplied by X. The power of four plus two X divided by, we execute minus X squared. If we substitute infinity for every X, we're going to get infinity in the numerator and infinity in the denominator, right. So essentially, since this is an indeterminate form, and we have a rational function, we want to divide each side of the rational function by the variable with the highest exponent. So that's X four, we end up with limit when X approaches infinity. And now we want to divide nine X fourth by X fourth which gives us nine. Then we are dividing two X by X fourth. This gives us two divided by X cubed. In the denominator, we X squared divided by X four gives us three, right. So we have three X cubed divided by X four. This is three divided by X and then we are subtracting X squared divided by X fourth which is one divided by X squared. Now let's go ahead and substitute infinity for every AX. We get nine plus two divided by infinity Q which is infinity. This approaches zero from the right. Then in the denominator, we have we divided by infinity which approaches zero from the right and we are subtracting one divided by infinity which approaches zero from the right. Now we end up with nine divided by zero from the right nine divided by zero from the right, gives us an infinitely large positive number. So we get positive infinity. Let's go ahead and evaluate the limit. When X approaches negative infinity, we can use our simplified expression nine plus two divided by X cubed. In the denominator, we have we divided by X minus one divided by X squared. So now we have mime plus two divided by negative infinity, right. So in this case, we are subtracting zero, right? Because we are approaching zero from the left, that's an infinitely small number. And now if we look at the denominator, we divided by X, when X approaches negative infinity is zero. Once again, we are approaching zero from the left and we are subtracting one divided by infinity squared, negative infinity squared which is infinity squared. So we are subtracting zero right. And in this case, we have nine in the numerator divided by zero from the left because we are subtracting zero from zero from the left. And in this case, nine divided by an infinitely small negative number gives us negative infinity. So now if we look at those limits, while we have our answers for each limit, and now we can state that the horizontal Asymptote will not exist because a horizontal Asymptote must have a form of Y equals A where A is a finite number. But each limit corresponds to either a positive or negative infinity. So we can state that the correct answer to this problem is option B limit one X approaches infinity of F FX is positive infinity. When X approaches negative infinity is negative infinity and there is no horizontal Asymptote. Thank you for watching.

Determine $\(\lim\)_{x\(\rightarrow\]\infty\)}f\(\left\)(x\(\right\))$ and $\(\lim\)_{x\(\rightarrow\)-\(\infty\)}f\(\left\)(x\(\right\))$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).

$f (x) = \frac{40 x^{5} + x^{2}}{16 x^{4} - 2 x}$

Key Concepts

Limits at Infinity

Horizontal Asymptotes

Rational Functions

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