Determine the following limits. lim x→∞ $$6x2/(4x^2$$+√(16x4 + x2))

Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 2 - Limits

Problema 32

Capitolo 2, Problema 32

Determine the following limits.

lim x→∞ 6x²/(4x^2+√(16x⁴ + x²))

Guida verificata passo dopo passo

Identify the dominant terms in the numerator and the denominator. In the numerator, the dominant term is $6x^2$. In the denominator, the dominant term is $4x^2$ and the term under the square root is $16x^4$.

Factor out the highest power of $x$ from both the numerator and the denominator. In this case, factor $x^2$ from both.

Rewrite the expression: $\frac{6x^2}{4x^2 + \sqrt{16x^4 + x^2}} = \frac{6}{4 + \sqrt{16 + \frac{1}{x^2}}}$.

As $x$ approaches infinity, the term $\frac{1}{x^2}$ approaches zero. Simplify the expression to $\frac{6}{4 + \sqrt{16}}$.

Calculate the simplified expression by evaluating the square root and performing the arithmetic operations.

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