Determine the following limits. lim x→∞ (3x12 − 9x7)

Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 2 - Limits

Problema 2.21

Capitolo 2, Problema 2.21

Determine the following limits.
lim x→∞ (3x¹² − 9x⁷)

Guida verificata passo dopo passo

Identify the highest power of x in the expression, which is x^{12}.

Factor out x^{12} from the expression: 3x^{12} - 9x^{7} = x^{12}(3 - 9x^{-5}).

Rewrite the limit expression: $ \lim_{x \to \infty} x^{12}(3 - 9x^{-5}) $.

Evaluate the limit of the factored expression: $ \lim_{x \to \infty} x^{12} $ and $ \lim_{x \to \infty} (3 - 9x^{-5}) $.

Since $ x^{12} \to \infty $ and $ 3 - 9x^{-5} \to 3 $, the overall limit is determined by the behavior of x^{12}.

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Determine $$\lim$_{x$\rightarrow$$\infty$}f$\left$(x$\right$)$ and $$\lim$_{x$\rightarrow$-$\infty$}f$\left$(x$\right$)$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).