AP Calculus AB and BC

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

Q: Does the function ever reach 1?

No. For positive x the denominator always exceeds the numerator, so the values stay just below 1 and rise toward it.

Q: What is the limit at negative infinity?

Also 1, so y = 1 is a horizontal asymptote in both directions.

The limit of x over x plus 1 as x approaches infinity is 1. The numerator and denominator have the same degree, so the limit is the ratio of the leading coefficients, which is 1 over 1. Dividing through by x makes it visible.

\lim_{x \to \infty} \frac{x}{x+1} = 1

Settled by dividing by the highest power.

Dividing by the highest power

Substitution gives $\frac{\infty}{\infty}$ . Dividing every term by $x$ , the highest power present, turns the unbounded parts into vanishing ones.

\frac{x}{x+1} = \frac{1}{1 + \frac{1}{x}}

\lim_{x \to \infty}\frac{1}{1+\frac{1}{x}} = \frac{1}{1+0} = 1

The degree rule

For a rational function: bottom-heavy gives 0, equal degrees give the ratio of leading coefficients, and top-heavy grows without bound. This one is the equal case.

Approaching from below, and the other end

The values are always slightly LESS than $1$ for positive $x$ , since the denominator is always the larger of the two. At $x = 100$ the value is about $0.990$ , so the graph rises toward the horizontal asymptote $y = 1$ without reaching it.

At negative infinity the limit is also $1$ , so $y = 1$ is a horizontal asymptote in both directions. There is a vertical asymptote at $x = -1$ , which is a separate feature and not what this limit describes.

The mistakes students make

Cancelling the $x$ to get $\frac{1}{1}$ by crossing out symbols. The $x$ in the denominator is part of a sum, so it is not a factor and cannot be cancelled.
Answering $0$ because $\frac{1}{x} \to 0$ . That is the correction term, not the whole expression.
Confusing the horizontal asymptote with the vertical one at $x = -1$ . Limits at infinity describe end behaviour only.

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 / (x + 1) as x approaches infinity?

It is $1$ .

Does the function ever reach 1?

No. For positive $x$ the denominator always exceeds the numerator, so the values stay just below $1$ and rise toward it.

What is the limit at negative infinity?

Also $1$ , so $y = 1$ is a horizontal asymptote in both directions.