Practice Edge Cases in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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?

Showing a random 20 of 50 problems.

Example 1

hard

A function

f:\emptyset \to B

is called what? How many such functions exist?

Example 2

medium

A program computes

\frac{a}{b-c}

. Which relationship between

b

and

c

is the edge case to guard against?

Example 3

easy

How many elements are in the empty set

\emptyset

Example 4

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 5

medium

An arithmetic sequence has first term

a_1=4

and common difference

d=0

. Find

a_{10}

and the sum

S_{10}

Example 6

easy

For the function

f(x)=\ln x

, identify the edge of the domain.

Example 7

hard

Solve

ax^2 + bx + c = 0

when

a=0

b=2

c=-6

. Why does the quadratic formula fail?

Example 8

easy

For

\sqrt{x}

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

Example 9

easy

\frac{0}{5}

defined? Is

\frac{5}{0}

defined?

Example 10

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 11

easy

The formula 'average

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

' breaks for which value of

n

Example 12

medium

A geometric sequence

a, ar, ar^2,\dots

converges when summed to infinity if and only if ___.

Example 13

hard

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

Example 14

medium

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

Example 15

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 16

medium

Solve

|x|=x

. List all valid

x

Example 17

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 18

easy

True or false: a formula proven for

n \ge 2

automatically holds for

n=0

and

n=1

Example 19

easy

For

f(x) = \sqrt{x}

, identify the edge case and state the domain.

Example 20

medium

Test the edge cases

n=0

and

n=1

for the formula

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

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Related Concepts

Assumptions Counterexample Limiting Cases