AP Calculus AB and BC

Limit of (x + sin x) / x at Infinity Is 1

The limit of x plus sin x, over x, as x approaches infinity is 1. Splitting the fraction gives 1 plus sin x over x, and that second piece is squeezed to 0 because sine stays bounded while the denominator grows.

lim⁡x→∞x+sin⁡xx=1\lim_{x \to \infty} \frac{x + \sin x}{x} = 1

Settled by splitting the fraction, then the squeeze theorem.

Splitting off the bounded part

Split the fraction so the well-behaved part separates from the oscillating one.

x+sin⁡xx=1+sin⁡xx\frac{x + \sin x}{x} = 1 + \frac{\sin x}{x}

The second term is bounded over unbounded, squeezed between −1x-\frac{1}{x} and 1x\frac{1}{x}, so it goes to 00.

lim⁡x→∞(1+sin⁡xx)=1+0=1\lim_{x \to \infty}\left(1 + \frac{\sin x}{x}\right) = 1 + 0 = 1

This is why L'Hopital must not be used here

The form is infinity over infinity, so the rule looks legal. But differentiating gives 1 plus cos x over 1, which has NO limit, and the rule is silent when the new limit fails to exist. It never says the original limit fails too.

A rule with a one-way door

L'Hopital's rule says that IF the limit of the ratio of derivatives exists, THEN the original limit equals it. It does not run backwards. Here the derivative ratio has no limit, and yet the original limit is a perfectly ordinary 11.

So a failed L'Hopital attempt is not evidence of anything. It means the tool was wrong for the problem, and you go back and use one that fits, which here is splitting the fraction.

The mistakes students make

  • Concluding the limit does not exist because L'Hopital produced 1+cos⁡x1 + \cos x. The rule cannot be read in that direction.
  • Answering that the limit does not exist because sin⁡x\sin x oscillates forever. It does, but its contribution is shrinking to nothing.
  • Cancelling the xx terms to get sin⁡x1\frac{\sin x}{1}. The xx upstairs is part of a sum, not a factor.

Not sure which technique a limit wants?

The Limit Method Chooser walks the decision from direct substitution through factoring, the conjugate, and L'Hopital, and says why each one applies or fails.

Frequently asked questions

What is the limit of (x + sin x) / x as x approaches infinity?

It is 11.

Why does L'Hopital's rule fail here?

It gives 1+cos⁡x1 + \cos x, which has no limit. The rule only transfers a limit that EXISTS, so an inconclusive result says nothing about the original.

Does the graph settle onto y = 1?

It oscillates around y=1y = 1 forever, but with amplitude 1x\frac{1}{x}, which shrinks to nothing.