Limiting Cases Examples in Math

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

Extreme values of a parameter (approaching zero, infinity, or a critical threshold) used to check formulas and reveal simplified behavior.

What happens when things get really big, really small, or reach boundaries?

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: Limiting cases test formulas and reveal asymptotic behavior.

Common stuck point: Taking a limit as a variable approaches a value is different from substituting that value directly — undefined expressions require careful limit analysis.

Sense of Study hint: Substitute a very large number (like 1000) and a very small number (like 0.001) into your formula. Compare the outputs to see what the expression approaches.

Worked Examples

Example 1

easy
The formula for the sum of a geometric series S = \frac{a(1-r^n)}{1-r} (r\ne 1). Find the limiting case as n \to \infty when |r| < 1.

Solution

  1. 1
    As n \to \infty with |r| < 1: r^n \to 0.
  2. 2
    Therefore S = \frac{a(1-r^n)}{1-r} \to \frac{a(1-0)}{1-r} = \frac{a}{1-r}.
  3. 3
    This is the sum of an infinite geometric series with |r|<1.

Answer

S_{\infty} = \frac{a}{1-r} \quad (|r|<1)
Taking a limiting case (n \to \infty) of a finite formula often yields a simpler infinite-series formula. The limit r^n \to 0 for |r|<1 is the key step.

Example 2

medium
The compound interest formula is A = P(1+r/n)^{nt}. Analyse the limiting cases n=1 (annual), n=12 (monthly), and n \to \infty (continuous).

Practice Problems

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

Example 1

easy
For the area of a regular n-gon inscribed in a circle of radius r: A_n = \frac{n}{2}r^2\sin(2\pi/n). What does A_n approach as n \to \infty?

Example 2

medium
The binomial (1+x/n)^n has a famous limiting case. Evaluate for n=1, 2, 10, 100 with x=1 and identify the limit.
← Back to Limiting Cases Formula Explained Practice Mode

Related Concepts

Edge CasesLimit

Background Knowledge

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

edge cases