What does a limit actually mean?

It is the value a function approaches as the input approaches some number — the trend, not the destination. The function need not be defined at that point, and if it is, the limit may still differ from it.

What do I do when I get \frac{0}{0}?

Treat it as a prompt, not an answer. Factor and cancel for polynomials, multiply by the conjugate when a radical is involved, or use L'Hôpital's rule if you have covered derivatives. Then substitute again.

When does a limit not exist?

When the two one-sided limits disagree (a jump), when the function grows without bound, or when it oscillates endlessly, as \sin\frac{1}{x} does near zero. An indeterminate form on its own is not one of these cases.

How do I find a limit at infinity?

For a rational function, compare degrees. Lower on top gives 0; equal degrees give the ratio of leading coefficients; higher on top gives \pm\infty. To show the working, divide every term by the highest power in the denominator.

What is the difference between a limit and continuity?

A limit only asks what the function approaches. Continuity at a requires three things: f(a) is defined, the limit exists, and the two are equal. A removable hole has a limit but is not continuous.

Why do limits matter for the rest of calculus?

Because the two central objects are defined as limits. The derivative is the limit of \frac{f(x+h)-f(x)}{h} as h \to 0, and the definite integral is the limit of a Riemann sum. Limits are the machinery underneath both.

You can start solving limits for free by adding the Chrome extension — try it on your own problem set before deciding if you need anything more.

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\lim_{x \to 3} (x^2 + 1)

1Polynomials are continuous everywhere, so substitute directly.
2 $3^2 + 1 = 10$ .

On this page01Definition 02Formulas 03Methods 04Examples 05Practice 06Mistakes 07FAQ

01Definition

What is a limit?

A limit describes what a function approaches as its input approaches some value — not necessarily what it equals there. $\lim_{x \to 2} f(x) = 5$ says that as $x$ gets arbitrarily close to $2$ , $f(x)$ gets arbitrarily close to $5$ , whether or not $f(2)$ is even defined.

That gap between "approaches" and "equals" is the entire point. $\frac{x^2 - 4}{x - 2}$ is undefined at $x = 2$ because the denominator is zero, yet the function settles on $4$ from both sides. The limit sees the trend; the value at the point is a separate question.

Limits are the foundation the rest of calculus sits on. A derivative is a limit of a difference quotient, a definite integral is a limit of a sum, and continuity is defined as the limit matching the function value. Everything downstream is built here.

Where you'll actually use this

Instantaneous speed: average speed over a shrinking time interval, in the limit, is the speedometer reading.
Compound interest compounded ever more frequently approaches $e$ — a limit.
Terminal velocity and other long-run steady states are limits as time goes to infinity.
Numerical methods and floating-point analysis, where convergence is a limit statement.
Marginal cost in economics: the limit of the cost of one more unit as the increment shrinks.

Limit: The value a function approaches as its input approaches a given point.
Indeterminate form: An expression like $\frac{0}{0}$ or $\frac{\infty}{\infty}$ that carries no information until the algebra is done.
One-sided limit: The value approached from only the left ( $x \to a^-$ ) or only the right ( $x \to a^+$ ).
Continuity: A function is continuous at $a$ when $f(a)$ exists, the limit exists, and they are equal.
Removable discontinuity: A single-point hole where the limit exists but the function is undefined — what cancelling a common factor reveals.

02Formulas

The formulas you need

Keep these on hand — every method below builds on them.

Notation

\lim_{x \to a} f(x) = L

As x approaches a, f(x) approaches L.

Two-sided existence

\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)

A limit exists only if both one-sided limits agree.

Indeterminate form

\frac{0}{0}

Not an answer — a signal that algebra is needed before substituting.

Limits at infinity

\lim_{x \to \infty} \frac{1}{x^n} = 0

For n > 0 — the workhorse behind every rational-function limit at infinity.

Continuity

\lim_{x \to a} f(x) = f(a)

The definition: the limit exists and matches the actual value.

Special trig limit

\lim_{x \to 0} \frac{\sin x}{x} = 1

x in radians. Worth memorising — it underlies the derivative of sine.

03Methods

How to solve limits

Pick the method that fits the problem in front of you.

When to use it: Always try this first. It works for every polynomial and for any rational function whose denominator is non-zero at the point.

1Substitute the value into the function: $\lim_{x \to 3}(x^2 + 1)$ gives $9 + 1 = 10$ .
2A defined real number is the answer — stop there.
3If you get $\frac{0}{0}$ , the limit still probably exists; move to an algebraic technique.
4If you get a non-zero number over zero, such as $\frac{5}{0}$ , the limit is infinite or does not exist.

04Worked examples

Limits, solved step by step

From easy to hard — pick a problem to see its full solution.

\lim_{x \to 3} (x^2 + 1)

1Polynomials are continuous everywhere, so substitute directly.
2 $3^2 + 1 = 10$ .

Try it on your own limits homework

Photo Math Solver reads a limit straight from a screenshot — a $\frac{0}{0}$ form that needs factoring, a radical needing a conjugate, a rational function heading to infinity — and shows the full working: what direct substitution gave, why that was not the answer, and which algebraic move resolved it.

05Practice

Now you try

Work each one on paper first, then check your answer.

Problem 1

\lim_{x \to 4} (3x - 5)

Problem 2

\lim_{x \to -3} \frac{x^2 - 9}{x + 3}

Problem 3

\lim_{x \to \infty} \frac{4x + 1}{x^2 - 2}

Problem 4

\lim_{x \to 5} \frac{x^2 - 25}{x - 5}

Problem 5

\lim_{x \to 0} \frac{\sin 3x}{x}

Stuck on your limits homework? Screenshot it

Limit notation stacks a subscript under the operator and often wraps a nested fraction or radical, which makes it slow to retype and easy to garble. Screenshot the whole expression with Photo Math Solver and the solution states what direct substitution gave, why it was inconclusive, and which technique resolved it.

06Common mistakes

Where points get lost

Each of these shows up on real graded work — and each has a simple fix.

The mistake	Why it happens	The fix
Reporting $\frac{0}{0}$ as the answer, or as "does not exist".	Substitution is the first technique taught, and whatever it produces looks like the result.	$\frac{0}{0}$ is indeterminate — it means the method was inconclusive, not that the limit fails. Factor, use a conjugate, or apply L'Hôpital's rule, then substitute again.
Assuming $\lim_{x \to a} f(x)$ must equal $f(a)$ .	It is true for every polynomial, which is where all the early examples come from.	That equality is the definition of continuity, not a general rule. A limit can exist where the function is undefined, and can differ from the value at a jump or a removable hole.
Refusing to cancel a factor because it is zero at the limit point.	Cancelling $(x - 2)$ when $x \to 2$ looks like dividing by zero, which is correctly drilled as forbidden.	The limit considers $x$ arbitrarily close to $2$ but never equal to it, so $(x - 2)$ is never actually zero. Cancelling is valid throughout the region the limit cares about.
Concluding a two-sided limit exists after checking only one side.	One side gives a clean number that looks like a finished answer.	Check both. For piecewise functions, absolute values, and vertical asymptotes, the sides routinely disagree — and if they do, the two-sided limit does not exist.
Treating $\infty$ as a number and writing $\frac{\infty}{\infty} = 1$ .	It appears in the same position an ordinary value would, so ordinary cancellation seems to apply.	$\frac{\infty}{\infty}$ is indeterminate, exactly like $\frac{0}{0}$ . Compare degrees, or divide through by the highest power, to get the real answer.

Photo Math Solver shows every intermediate step, so slips like these are easy to catch before they cost you marks.

07FAQ