Let L=lim _x→∞(x+sin x)/(x). With f=x+sin x, g=x, the quotient is (f')/(g')=1+cos x. What is true?

The limit is 1, even though f'/g' has no limit

The limit does not exist, because 1+cos x has no limit

Exercise · ZW-LHOPITALS-RULE-020 · ★★★ Advanced

Let \(\displaystyle L=\lim_{x\to\infty}\frac{x+\sin x}{x}\). With \(f=x+\sin x\), \(g=x\), the quotient is \(\frac{f'}{g'}=1+\cos x\). What is true?

Choose one option.

Show answer

C. The limit is 1, even though f'/g' has no limit

Explanation

\(\frac{x+\sin x}{x}=1+\frac{\sin x}{x}\to1\). The rule only guarantees equality if the limit of \(\frac{f'}{g'}\) exists; if it does not, it says nothing.

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Review the concept: L'Hôpital's rule

Rule that computes limits of quotients in indeterminate form \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\) by differentiating numerator and denominator separately.

\[\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f'(x)}{g'(x)}\]

Let \(\displaystyle L=\lim_{x\to\infty}\frac{x+\sin x}{x}\). With \(f=x+\sin x\), \(g=x\), the quotient is \(\frac{f'}{g'}=1+\cos x\). What is true?

Explanation

Review the concept: L'Hôpital's rule

More practice problems: l'hôpital's rule