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=0x = 0? When x→∞x \to \infty? When inputs are unusual?

Showing a random 20 of 50 problems.

Example 1

hard
A function f:∅→Bf:\emptyset \to B is called what? How many such functions exist?

Example 2

medium
A program computes ab−c\frac{a}{b-c}. Which relationship between bb and cc 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 nn people is (n2)=n(n−1)2\binom{n}{2}=\frac{n(n-1)}{2}. Verify the edge cases n=0n=0 and n=1n=1.

Example 5

medium
An arithmetic sequence has first term a1=4a_1=4 and common difference d=0d=0. Find a10a_{10} and the sum S10S_{10}.

Example 6

easy
For the function f(x)=ln⁡xf(x)=\ln x, identify the edge of the domain.

Example 7

hard
Solve ax2+bx+c=0ax^2 + bx + c = 0 when a=0a=0, b=2b=2, c=−6c=-6. Why does the quadratic formula fail?

Example 8

easy
For x\sqrt{x} over the reals, what range of inputs is an edge/invalid case?

Example 9

easy
Is 05\frac{0}{5} defined? Is 50\frac{5}{0} defined?

Example 10

challenge
A binary search on a sorted array uses midpoint mid=low+high2\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 =sumn= \frac{\text{sum}}{n}' breaks for which value of nn?

Example 12

medium
A geometric sequence a,ar,ar2,…a, ar, ar^2,\dots converges when summed to infinity if and only if ___.

Example 13

hard
lim⁡x→∞(1+1/x)x=?\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⋅f(n−1)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|x|=x. List all valid xx.

Example 17

medium
lim⁡x→0sin⁡xx\lim_{x\to 0}\frac{\sin x}{x} has the edge form 0/00/0 at x=0x=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≥2n \ge 2 automatically holds for n=0n=0 and n=1n=1.

Example 19

easy
For f(x)=xf(x) = \sqrt{x}, identify the edge case and state the domain.

Example 20

medium
Test the edge cases n=0n=0 and n=1n=1 for the formula ∑k=1nk=n(n+1)2\sum_{k=1}^{n} k = \frac{n(n+1)}{2}.
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