Limit of a Function as x Approaches a

What a Limit Means

Look at $f(x) = \dfrac{x^2-1}{x-1}$ near $x = 1$. You cannot plug in $x = 1$, because the denominator would be $0$. But you can get as close to $1$ as you like, from the left and from the right. For $x \neq 1$, $f(x) = \dfrac{(x-1)(x+1)}{x-1} = x + 1$, so the graph is the line $y = x + 1$ with a hole at $(1, 2)$.

On a graph, an open circle marks a point that is not part of the graph, and a filled dot marks the actual value. The limit is the height the curve is heading toward, which you find by following the curve from both sides.

Direct Substitution

Most of the time you do not need tables. If $f$ is a nice function (a polynomial, a rational function, a radical, $e^x$, $\ln x$, $\sin x$, $\cos x$) and $a$ is in its domain, the curve has no break at $a$. So the limit is just the value:

The graph above has no break at $x = 2$, so the limit there is just the height of the curve.

One-Sided Limits

Sometimes the left side and the right side head to different places. That happens with piecewise functions and with absolute values, so you keep the two sides separate.

Here $\lim_{x\to 1^-} f(x) = 3$ and $\lim_{x\to 1^+} f(x) = 1$. The sides disagree, so $\lim_{x\to 1} f(x)$ does not exist.

Resolving 0/0 by Factoring

When a rational function gives $\frac{0}{0}$ at $x = a$, both the numerator and the denominator have the factor $(x - a)$. Because the limit never uses $x = a$ itself, you may cancel that factor and then substitute.

The graph is the line $y = x + 2$ with a hole at $x = 3$. The limit is the height of the hole, $5$.

Cubics need a root first. If the numerator is $0$ at $x = a$, then $(x - a)$ is a factor, and you can find the other factor with synthetic division. Some problems call this Ruffini's rule: it is the same method.

Other useful identities: $a^2 - b^2 = (a-b)(a+b)$ and $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$.

Radicals: Multiply by the Conjugate

When the $\frac{0}{0}$ involves a square root, factoring does not help. Instead, multiply the top and bottom by the conjugate of the radical expression. The conjugate of $\sqrt{u} - c$ is $\sqrt{u} + c$. The product is a difference of squares that removes the root:

Tougher 0/0 Forms

Three more tools handle the harder cases.

Cubics. Find the root with substitution, divide with synthetic division, and factor the quotient. You may need to do this twice if $(x - a)$ appears as a double factor.

Radicals in the numerator and the denominator. Rationalize each one with its own conjugate. The zero factor that appears in both cancels.

Different roots. When square roots and cube roots (or other indices) mix, substitute $x = t^n$, where $n$ is the least common multiple of the indices. Then every root becomes a power of $t$. Remember that $x \to a$ means $t \to a^{1/n}$.

Infinite Limits and Vertical Asymptotes

If direct substitution gives $\frac{k}{0}$ with $k \neq 0$, the denominator is tiny near $a$, so the quotient gets huge. It heads to $+\infty$ or $-\infty$, and the sign depends on each side.

Be careful: a factor that is not a pure zero-denominator still needs the sign of the whole denominator, so factor it when you can. Absolute values also need a side-by-side look, as in this example.

Reading Limits from Graphs and Combining Them

Reading a limit from a graph means tracing the curve with your finger toward $x = a$ from each side and noting the height you are heading to.

• A filled dot is the real value $f(a)$. An open circle is a point the graph skips.

• If your two fingers meet at the same height, the limit exists, no matter what is plotted at $x = a$.

• If they land at different heights (a jump), the limit does not exist, but each one-sided limit does.

• If the curve shoots up or down along a vertical line, the limit is $\pm\infty$.

When a function is built from others, the limit laws let you combine one-sided limits that you read from graphs.

In the picture, $h(x) = x$ tends to $0$ at $x = 0$, and the reciprocal $\frac1x$ blows up in opposite directions, since $x$ changes sign.

Finding Unknown Parameters

Often a problem says "find the value of $k$ (or $a$, $b$, $m$, $n$) so that the limit exists or has a given value." Use these rules:

• Piecewise functions: set the two one-sided limits equal at each break point.

• A $\frac{0}{0}$ quotient with a finite limit: the denominator is $0$ at $a$, so the numerator must also be $0$ at $a$. That gives an equation for the parameter. Then factor and finish.

• Radical limits with a parameter: rationalize first, then solve the resulting equation for the parameter.

Drag $c$: the graph has a vertical asymptote at $x = -1$ unless $c$ takes one special value, where the asymptote turns into a hole. That value is the one that makes the limit exist.

Limits and Instantaneous Rate of Change

Back to the skateboarder. Suppose her position is $s(t) = t^2 + 3t$ feet. Her average speed from $t = 1$ to a nearby time $t$ is $\dfrac{s(t) - s(1)}{t - 1}$. That works for any $t \neq 1$. The instantaneous velocity at $t = 1$ is what these average speeds approach as $t \to 1$. It is a limit of $\frac{0}{0}$ type that you solve by factoring.

The curve above shows the average speed between $1$ and $x$. It has a hole at $x = 1$, and the height of the hole is the instantaneous velocity.

This limit is called the derivative of $s$ at $t = 1$. You will use exactly this idea in calculus.

Summary