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Math  /  Calculus

Questionlim⁡x→−∞x4−94x2−16=\lim _{x \rightarrow-\infty} \frac{x^{4}-9}{4 x^{2}-16}=

Studdy Solution

STEP 1

1. We are dealing with a limit as x x approaches negative infinity.
2. The expression is a rational function, which is a ratio of two polynomials.
3. We will simplify the expression to find the behavior as x→−∞ x \to -\infty .

STEP 2

1. Identify the leading terms in the numerator and denominator.
2. Simplify the expression by dividing by the highest power of x x in the denominator.
3. Evaluate the limit of the simplified expression as x→−∞ x \to -\infty .

STEP 3

Identify the leading terms in the numerator and the denominator. The leading term in the numerator is x4 x^4 and in the denominator is 4x2 4x^2 .

STEP 4

Divide every term in the numerator and the denominator by x2 x^2 , the highest power of x x in the denominator:
x4−94x2−16=x4x2−9x24x2x2−16x2\frac{x^4 - 9}{4x^2 - 16} = \frac{\frac{x^4}{x^2} - \frac{9}{x^2}}{\frac{4x^2}{x^2} - \frac{16}{x^2}}
This simplifies to:
x2−9x24−16x2\frac{x^2 - \frac{9}{x^2}}{4 - \frac{16}{x^2}}

STEP 5

Evaluate the limit of the simplified expression as x→−∞ x \to -\infty . As x→−∞ x \to -\infty , the terms 9x2\frac{9}{x^2} and 16x2\frac{16}{x^2} approach 0:
lim⁡x→−∞x2−9x24−16x2=x24\lim_{x \to -\infty} \frac{x^2 - \frac{9}{x^2}}{4 - \frac{16}{x^2}} = \frac{x^2}{4}
Since x2 x^2 grows without bound as x→−∞ x \to -\infty , the limit is:
lim⁡x→−∞x24=∞\lim_{x \to -\infty} \frac{x^2}{4} = \infty
However, since we are considering x→−∞ x \to -\infty , the behavior of x2 x^2 is positive, and the overall limit is positive infinity.
The value of the limit is:
∞\boxed{\infty}

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