What is lim_(x rarr oo)(x^(2)-4)/(2+x-4x^(2))? (A) -2 (B) -(1)/(4) (C) (1)/(2) (D) 1 (E) The limit does not exist.

Identify Power Levels: Identify the highest power of x in the numerator and the denominator to simplify the expression. Numerator highest power: x^2 Denominator highest power: -4x^2Simplify by Division: Simplify the expression by dividing both the numerator and the denominator by x^2. lim_(x rarr oo) [(x^2/x^2) - (4/x^2)] / [(2/x^2) + (x/x^2) - (4x^2/x^2)] = lim_(x rarr oo) [1 - 0] / [0 + 0 - 4] = lim_(x rarr oo) 1 / -4Calculate Final Limit: Calculate the final limit. lim_(x rarr oo) 1 / -4 = -1/4

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Calculus

Find derivatives of logarithmic functions

Full solution

Q. What is

\lim _{x \rightarrow \infty} \frac{x^{2}-4}{2+x-4 x^{2}} ?

\newline

(A)

-2

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(B)

-\frac{1}{4}

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(C)

\frac{1}{2}

\newline

(D)

1

\newline

(E) The limit does not exist.

Identify Power Levels: Identify the highest power of $x$ in the numerator and the denominator to simplify the expression. $\newline$ Numerator highest power: $x^2$ $\newline$ Denominator highest power: $-4x^2$
Simplify by Division: Simplify the expression by dividing both the numerator and the denominator by $x^2$ . $\newline$ $\lim_{x \to \infty} \left[\frac{x^2}{x^2} - \frac{4}{x^2}\right] / \left[\frac{2}{x^2} + \frac{x}{x^2} - \frac{4x^2}{x^2}\right]$ $\newline$ $= \lim_{x \to \infty} \left[1 - 0\right] / \left[0 + 0 - 4\right]$ $\newline$ $= \lim_{x \to \infty} \frac{1}{-4}$
Calculate Final Limit: Calculate the final limit. $\lim_{x \rightarrow \infty} \frac{1}{-4} = -\frac{1}{4}$