Understanding Limits

Calculus is the math of change. One half measures how fast something is changing at a single instant, and the other half adds up small amounts to give a total.

Understanding Limits: the central idea

A limit describes the output approached as the input moves near a target. The function may have a different value—or no value—at the target itself.

Words you need

Limit

A limit is the single number a function's output moves toward as the input gets closer and closer to a chosen value.

One-sided limit

A one-sided limit is the value a function heads toward when you approach the target from only the left side or only the right side.

Indeterminate form

An indeterminate form such as $0/0$ or $\infty/\infty$ is a result that gives no information yet, so the expression must be rewritten before the limit can be read.

Removable discontinuity

A removable discontinuity is a single missing point, or hole, in a graph where the limit exists but the function has no value there.

Continuity

Continuity at a point means the limit exists there, the function is defined there, and those two numbers are equal, so the graph can be drawn through the point without lifting your pencil.

Squeeze theorem

The squeeze theorem says that if one function is trapped between two others that both head to the same number, the trapped function must head to that number too.

What to know before this lesson

Be able to evaluate functions, factor expressions, read graphs, and distinguish a point value from nearby behavior.

If one of those prerequisites is uncertain, use the Calculus subject guide to locate the earlier concept before memorizing a procedure.

Limits: a worked example

Every step, with the arithmetic

1. Step 1 - Write the limit you want$\lim_{x\to3}\dfrac{x^2-9}{x-3}$
2. Step 2 - Try plugging in 3 first$(3^2-9)/(3-3) = 0/0$, which is an indeterminate form, not an answer
3. Step 3 - Factor the top$x^2-9=(x-3)(x+3)$, so the quotient is $(x-3)(x+3)/(x-3)$
4. Step 4 - Cancel the matching factorFor every $x$ except $3$, the quotient equals $x+3$
5. Step 5 - Check from the left$x=2.9 \Rightarrow 2.9+3=5.9$; $x=2.99 \Rightarrow 5.99$
6. Step 6 - Check from the right$x=3.1 \Rightarrow 6.1$; $x=3.001 \Rightarrow 6.001$
7. Step 7 - Both sides agree, so the limit exists$\lim_{x\to3}\dfrac{x^2-9}{x-3}=3+3=6$, even though the original fraction is undefined at $x=3$

For $(x^2-1)/(x-1)$, direct substitution gives $0/0$. Factoring and cancelling for $x\ne1$ leaves $x+1$, whose nearby values approach $2$.

Limits worth memorizing, with the value each one approaches

The step-by-step method for limits

1. Try direct substitution first and identify whether it gives a number or an indeterminate form.
2. Use algebra, a graph, a table, or one-sided analysis to expose nearby behavior without changing the limit.
3. Compare left- and right-hand limits; a two-sided limit exists only when they agree.

How to check your answer

Evaluate at values such as $0.99$ and $1.01$, and verify that both sides move toward $2$ even though the original quotient is undefined at $1$.

Compare units and qualitative behavior. A derivative should match the graph's slope, while a definite integral should agree with the signed area and scale of the interval.

A mistake that changes the mathematics

The expression $0/0$ is not a limit value. It is a signal that more analysis is required.

Where you will actually use this

Instantaneous speed

A car's speedometer reading is a limit. Average speed over shorter and shorter time windows, such as 1 second then 0.1 second then 0.01 second, settles on the speed at that exact instant.

Compound interest

Interest paid yearly, then monthly, then daily, then every second gives growth factors that close in on the limit $(1+1/n)^n = e$, which is why continuous compounding uses $e$.

Computer graphics and zoom

Rendering a smooth curve means taking the limit of many short straight segments, so the picture looks curved no matter how far a user zooms in.

How limits connects to the rest of calculus

• Derivative rules — A derivative is literally a limit of slopes, so every derivative rule is a shortcut for a limit you would otherwise compute by hand.
• Factoring polynomials — Most $0/0$ limits are fixed by factoring the top and bottom and cancelling the shared factor.
• Fundamental theorem of calculus — A definite integral is defined as the limit of a sum of thin rectangles, so limits sit underneath integration too.

Try a transfer problem

Find $\lim_{x\to4}(\sqrt{x}-2)/(x-4)$ by rationalizing, then explain why cancellation does not redefine the original function at $4$.

Show the worked answer

Multiply the top and bottom by the conjugate $\sqrt{x}+2$. The top becomes $(\sqrt{x}-2)(\sqrt{x}+2)=x-4$, so the fraction is $(x-4)/[(x-4)(\sqrt{x}+2)]$. For every $x$ near $4$ but not equal to $4$, the $(x-4)$ factors cancel and leave $1/(\sqrt{x}+2)$. Substituting $4$ now gives $1/(2+2)=1/4$, so the limit is $\tfrac14=0.25$. Cancelling did not redefine the original function: at $x=4$ the first expression is still $0/0$ and still undefined. The cancellation is only legal because a limit never looks at the target point itself, only at the points around it. The graph of the original quotient is the graph of $1/(\sqrt{x}+2)$ with a hole punched out at $(4,0.25)$.

Work without copying the example. When finished, use the relevant focused calculator or formula reference to check the setup and result, then correct the first line where your reasoning changed. When the method feels reliable, move to calculus practice questions.

Questions about limits

Can a limit exist at a point where the function is undefined?

Yes, and that is the most common case. In $(x^2-9)/(x-3)$ the value at $x=3$ does not exist, but the outputs on both sides march toward $6$, so the limit is $6$. A limit only cares about the neighborhood, never the point itself.

What is the difference between a limit being infinite and a limit not existing?

Both mean there is no finite answer. Writing the limit as $\infty$ is extra information: it says the outputs grow without bound in one clear direction. If the left side heads to $-\infty$ and the right side to $+\infty$, or the values keep oscillating, you just say the limit does not exist.

How do I find a limit as $x$ goes to infinity for a fraction of polynomials?

Compare the highest powers. If the top and bottom have the same highest power, the limit is the ratio of those leading coefficients, so $(3x^2+5)/(6x^2-x)$ goes to $3/6=0.5$. If the bottom's power is bigger the limit is $0$, and if the top's power is bigger there is no finite limit.

Why is a table of nearby values not a proof?

A table only shows a handful of inputs. A function can look calm at $x=0.1, 0.01, 0.001$ and still misbehave between those values, which happens with things like $\sin(1/x)$. Tables build intuition; algebra, the squeeze theorem, or the formal definition supply the proof.