Edge Cases Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Edge Cases.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Special or extreme input values — such as zero, infinity, empty sets, or boundary conditions — where formulas or reasoning may behave differently.

What happens at the extremes? When $x = 0$ ? When $x \to \infty$ ? When inputs are unusual?

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Edge cases are the extreme or special inputs — zero, infinity, empty, the boundary — where a formula or argument can behave differently.

Common stuck point: The procedure for edge cases is the easy part; the trap is testing only typical inputs. Asking "Am I deliberately testing the extreme or special inputs where a formula or argument might behave differently?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I deliberately testing the extreme or special inputs where a formula or argument might behave differently?

Worked Examples

Example 1

easy

For the function

f(x) = \dfrac{x^2-4}{x-2}

, check the edge case

x = 2

and describe what happens.

Answer

f(x) = x+2 \text{ for } x \ne 2;\quad f(2) \text{ is undefined (hole at }(2,4)\text{)}

First step

x = 2

: the denominator

x - 2 = 0

, so

f(2)

is undefined — this is the edge case.

Full solution

2
For $x \ne 2$ : factor the numerator — $\frac{x^2-4}{x-2} = \frac{(x-2)(x+2)}{x-2} = x + 2$ .
3
So $f(x) = x+2$ for all $x \ne 2$ . There is a hole in the graph at $x=2$ , $y=4$ .

Edge cases are special inputs where a formula breaks or behaves differently. Checking

x = 2

(where the denominator vanishes) is essential for understanding the full behaviour of

f

Example 2

medium

Check all edge cases for the statement: 'For natural numbers

n

\dfrac{n!}{(n-1)!} = n

.' Test

n = 0

and

n = 1

Example 3

medium

The formula

\binom{n}{k}=\frac{n!}{k!(n-k)!}

uses

0!=1

. Verify

\binom{5}{0}=1

and

\binom{5}{5}=1

using this convention.

Example 4

medium

Solve

|x|=x

. List all valid

x

Example 5

hard

The discriminant of

x^2+bx+1=0

vanishes for what

b

? What happens to the roots?

Example 6

challenge

Prove by checking edges that

\sum_{k=0}^{n}\binom{n}{k}=2^n

holds at

n=0

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

For

f(x) = \sqrt{x}

, identify the edge case and state the domain.

Example 2

medium

Test the edge cases

n=0

and

n=1

for the formula

\sum_{k=1}^{n} k = \frac{n(n+1)}{2}

Example 3

easy

For the expression

\frac{1}{x}

, which input value breaks it?

Example 4

easy

The formula 'average

= \frac{\text{sum}}{n}

' breaks for which value of

n

Example 5

easy

For

\sqrt{x}

over the reals, what range of inputs is an edge/invalid case?

Example 6

easy

A loop runs 'for

i

from

1

n

'. What value of

n

makes it run zero times?

Example 7

easy

The slope formula

\frac{y_2-y_1}{x_2-x_1}

breaks when which condition holds?

Example 8

easy

True or false: a formula proven for

n \ge 2

automatically holds for

n=0

and

n=1

Example 9

easy

\tan\theta

is undefined at which standard angle in

[0,\pi)

Example 10

easy

The quadratic formula uses

\frac{-b\pm\sqrt{b^2-4ac}}{2a}

. What value of

a

breaks it?

Example 11

medium

Solving

\frac{x}{x-2} = 3

, a student gets

x=3

. Why must they still check the edge case, and is

x=3

valid?

Example 12

medium

The formula for the sum of a geometric series

\frac{a(1-r^n)}{1-r}

breaks at which

r

, and what is the sum there?

Example 13

medium

A program computes

\frac{a}{b-c}

. Which relationship between

b

and

c

is the edge case to guard against?

Example 14

medium

The number of handshakes among

n

people is

\binom{n}{2}=\frac{n(n-1)}{2}

. Verify the edge cases

n=0

and

n=1

Example 15

medium

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

has the edge form

0/0

x=0

. What is the limit, and why isn't the function value the answer?

Example 16

medium

A median-finding routine assumes a sorted list. What edge case input could still produce a wrong/undefined result?

Example 17

medium

\log_b(x)

, besides

x>0

, what edge cases on the base

b

are invalid?

Example 18

medium

Cancelling

(x-3)

from both sides of

(x-3)(x+1)=(x-3)\cdot 4

can lose a solution. Which value is at risk and why?

Example 19

medium

The function

f(x)=\frac{|x|}{x}

is asked at every real input. Identify the edge case and the values on either side.

Example 20

challenge

The formula

\frac{x^2-a^2}{x-a}

simplifies to

x+a

. Identify the edge case, evaluate the limit there, and explain the removable discontinuity.

Example 21

challenge

A recursive factorial

f(n)=n \cdot f(n-1)

needs a base case to terminate. What is the edge case, and what value must it return?

Example 22

challenge

A binary search on a sorted array uses midpoint

\text{mid}=\frac{\text{low}+\text{high}}{2}

. Name the edge case that causes incorrect behavior on huge arrays and the fix.

Example 23

easy

What is

0!

Example 24

easy

What is the empty product

\prod_{i=1}^{0} a_i

Example 25

easy

Find the domain of

f(x) = \frac{1}{x-3}

Example 26

easy

How many elements are in the empty set

\emptyset

Example 27

easy

For the function

f(x)=\ln x

, identify the edge of the domain.

Example 28

medium

Solve

\frac{x^2-9}{x-3}=0

, listing all values that must be excluded.

Example 29

medium

Compute

\lim_{x\to 0}\frac{1-\cos x}{x^2}

Example 30

medium

For what value of

a

does

ax+3=0

have no solution?

Example 31

medium

An expression

\sqrt{x-5}+\sqrt{8-x}

has what domain?

Example 32

medium

Solve

x^2 + 1 = 0

over the reals. State the edge issue.

Example 33

medium

An arithmetic sequence has first term

a_1=4

and common difference

d=0

. Find

a_{10}

and the sum

S_{10}

Example 34

medium

A triangle has sides

3, 4, 7

. Does it exist? Cite the rule.

Example 35

medium

How many subsets does the empty set

\emptyset

have?

Example 36

hard

A function

f:\emptyset \to B

is called what? How many such functions exist?

Example 37

hard

Solve

ax^2 + bx + c = 0

when

a=0

b=2

c=-6

. Why does the quadratic formula fail?

Example 38

hard

\lim_{x\to\infty}(1+1/x)^x = ?

Example 39

hard

Solve

\sqrt{x+3} = x - 3

. Check edge cases for extraneous roots.

Example 40

challenge

Does a Pythagorean-style triangle exist with legs

a=b=0

and hypotenuse

c=0

? Comment on degeneracy.

← Back to Edge Cases Formula Explained Practice Mode

Related Concepts

Assumptions Counterexample Limiting Cases

Background Knowledge

These ideas may be useful before you work through the harder examples.

assumptions