QuestionFind $a$ so that $\lim_{{x \rightarrow \infty}} \frac{ax^{3}+4x^{2}-x-1}{3x^{3}+x^{2}-9} = -5$ .

Studdy Solution

STEP 1

Assumptions1. We are given the limit as $x$ approaches infinity of the function $\frac{ax^{3}+4x^{}-x-1}{3x^{3}+x^{}-9}$ . . We are told that this limit equals $-5$ .
3. We need to find the value of $a$ that makes this true.

STEP 2

When dealing with limits at infinity of rational functions (a ratio of two polynomials), we can use the rule that the limit is equal to the ratio of the leading coefficients if the degrees of the polynomials in the numerator and the denominator are the same.
In our case, the degree of the polynomials in the numerator and the denominator is, so we can apply this rule.
$\displaystyle\lim{{x \rightarrow \infty}} \frac{ax^{}+4x^{2}-x-1}{x^{}+x^{2}-9} = \frac{a}{}$

STEP 3

We are given that this limit equals $-5$ . So, we can set up the equation $\frac{a}{3} = -5$ to find the value of $a$ .
$\frac{a}{3} = -5$

STEP 4

To solve for $a$ , we can multiply both sides of the equation by3.
$a = - \times3$

STEP 5

Calculate the value of $a$ .
$a = -5 \times3 = -15$ So, the value of $a$ that makes the given limit equal to $-5$ is $-15$ .

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