Question

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.f(x) = (√(16x4 + 64x2) + x2) / (2x2 − 4)

Accepted Answer

Step 1: Simplify the expression inside the square root. Notice that the dominant term inside the square root as x approaches infinity is $$16x^4. $$Factor out $$x^4$$ from the square root: $$ \sqrt{16x^4 + 64x^2} = x^2\sqrt{16 + \frac{64}{x^2}} . $$Step 2: Simplify the expression for f(x) by dividing the numerator and the denominator by $$x^2$$, the highest power of x in the denominator: $$ f(x) = \frac{x^2\sqrt{16 + \frac{64}{x^2}} + x^2}{2x^2 - 4} = \frac{x^2(\sqrt{16 + \frac{64}{x^2}} + 1)}{2x^2 - 4} . $$Step 3: Evaluate the limit as x approaches infinity. As x approaches infinity, $$ \frac{64}{x^2} $$ approaches 0, so $$ \sqrt{16 + \frac{64}{x^2}} $$ approaches $$ \sqrt{16} = 4 . $$Therefore, the expression simplifies to $$ \frac{x^2(4 + 1)}{2x^2 - 4} = \frac{5x^2}{2x^2 - 4} . $$Step 4: Simplify the expression $$ \frac{5x^2}{2x^2 - 4}  by $$dividing the numerator and the denominator by $$x^2$$: $$ \frac{5}{2 - \frac{4}{x^2}} . As x $$approaches infinity, $$ \frac{4}{x^2} $$ approaches 0, so the expression approaches $$ \frac{5}{2} . $$Step 5: Evaluate the limit as x approaches negative infinity. The process is similar to the positive infinity case, and the expression $$ \frac{5}{2 - \frac{4}{x^2}} $$ also approaches $$ \frac{5}{2} . $$Therefore, the horizontal asymptote is y = \frac{5}{2}.

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.
f(x) = (√(16x⁴ + 64x²) + x²) / (2x² − 4)

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Key Concepts

Limits at Infinity

Horizontal Asymptotes

Rational Functions