Limiting Cases Examples: 47 Problems with Answers

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 a formula by sending a parameter to zero, infinity, or a critical threshold to reveal simplified behavior.

Common stuck point: The procedure for limiting cases is the easy part; the trap is trusting a formula from a middle value alone. Asking "Does the formula still make sense when I push a parameter to zero, infinity, or its critical value?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does the formula still make sense when I push a parameter to zero, infinity, or its critical value?

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

Answer

S_{\infty} = \frac{a}{1 - r} (∣ r ∣ < 1)

First step

n \to \infty

with

|r| < 1

r^n \to 0

Full solution

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

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).

Example 3

medium

Compound interest

A = P(1 + r/n)^{nt}

. Check the limiting case

r\to 0

. What does

A

approach?

Example 4

medium

Sanity-check the formula for a cone's volume

V = \tfrac13 \pi r^2 h

in the limit

r\to 0

(fixed

h

Example 5

medium

Sanity-check Pythagoras: a right triangle with legs

a

and

b\to 0

. What does the hypotenuse approach?

Example 6

hard

Check the quadratic formula by computing

x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}

c\to 0

. What roots do you expect?

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.

Example 3

easy

As x grows without bound, what value does 1/x approach?

Example 4

easy

What does the area formula A = pi r^2 give as r approaches 0?

Example 5

easy

As x approaches infinity, what does e^(-x) approach?

Example 6

easy

In the formula v = d/t, what happens to v as t approaches 0 (for fixed d > 0)?

Example 7

easy

What does (1/2)^n approach as n grows large?

Example 8

easy

A formula for resistance gives R as temperature T approaches absolute zero, modeled R = T/(T+1). What does R approach as T -> 0?

Example 9

easy

As n approaches infinity, what does n/(n+1) approach?

Example 10

easy

Sanity check: a population model gives P = 100/(1 + e^(-t)). What does P approach as t -> infinity?

Example 11

medium

Evaluate the limit of (x^2 - 1)/(x - 1) as x approaches 1.

Example 12

medium

A small-angle approximation uses sin(theta) approx theta for small theta. Use the limiting case to find the limit of sin(theta)/theta as theta -> 0.

Example 13

medium

Check the quadratic formula x = (-b +/- sqrt(b^2 - 4ac))/(2a) in the limit a -> 0. What should happen, and does the formula behave well?

Example 14

medium

A series sum is S = sum_{k=0}^{n} (1/2)^k. Find the limit of S as n -> infinity.

Example 15

medium

Test the formula for the sum 1 + 2 + ... + n = n(n+1)/2 by checking the limiting case n = 1. Does it pass?

Example 16

medium

As n -> infinity, what does (1 + 1/n)^n approach?

Example 17

challenge

Examine f(x) = (1 - cos x)/x^2 as x -> 0. Find the limit and explain why naive substitution fails.

Example 18

challenge

A formula for the area of a regular n-gon inscribed in a unit circle is A(n) = (n/2) sin(2*pi/n). Find the limit as n -> infinity and interpret it.

Example 19

challenge

Consider g(x) = x^x as x -> 0+ (x positive). Determine the limit and justify it.

Example 20

medium

Check the formula for resistors in parallel, R = (R1*R2)/(R1+R2), in the limit R2 -> infinity. What should R approach physically?

Example 21

medium

As x -> 0+, what does x*ln(x) approach? Use the dominance of x over ln(x).

Example 22

medium

Check the average-speed formula for a round trip at speeds u and v: avg = 2uv/(u+v). In the limit v -> 0 (return trip stalls), what does avg approach?

Example 23

easy

What does

\dfrac{1}{n^2}

approach as

n\to\infty

Example 24

easy

For the perimeter of a regular

n

-gon inscribed in a unit circle,

P_n = 2n\sin(\pi/n)

, what does

P_n

approach as

n\to\infty

Example 25

easy

Check the formula for cylinder volume

V=\pi r^2 h

in the limit

h\to 0

. What should

V

approach?

Example 26

easy

n \to \infty

, what does

\dfrac{2n+1}{n+3}

approach?

Example 27

easy

h\to 0

, what does

\dfrac{f(x+h) - f(x)}{h}

become, in general, for differentiable

f

Example 28

medium

Evaluate

\displaystyle\lim_{x\to 0}\dfrac{\tan x}{x}

Example 29

medium

For the partial sum

S_n = 1 + 1/2 + 1/4 + \ldots + 1/2^{n-1}

, what does

S_n

approach as

n\to\infty

Example 30

medium

n\to\infty

, what does

\sqrt[n]{n}

approach?

Example 31

medium

In the ideal gas law

PV = nRT

, what does pressure

P

approach as volume

V\to\infty

at fixed

T, n

Example 32

medium

Population grows as

P(t) = P_0 e^{kt}

with

k>0

. What does

P

approach as

t\to\infty

Example 33

medium

Logistic model

P(t) = \dfrac{K}{1 + Ae^{-rt}}

with

r>0

. What does

P(t)

approach as

t\to\infty

Example 34

medium

What does

\dfrac{\sin x}{x}

approach as

x\to\infty

Example 35

hard

Find

\displaystyle\lim_{x\to\infty}\left(\dfrac{x+1}{x-1}\right)^x

Example 36

hard

What does

n!

behave like for large

n

(Stirling's leading behavior)?

Example 37

hard

A relativistic energy formula gives

E = mc^2/\sqrt{1 - v^2/c^2}

. What does

E

approach as

v\to 0

Example 38

hard

For

f(x) = \dfrac{x^2 - 4}{x^2 - 5x + 6}

, evaluate

\lim_{x\to 2} f(x)

Example 39

hard

Consider

g(x) = (1+x)^{1/x}

for

x > 0

. What does

g(x)

approach as

x \to 0^+

Example 40

challenge

For the Binomial probability with

n

large and

p = \lambda/n

, find the limit of

P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}

Example 41

challenge

For lenses,

\dfrac{1}{f} = \dfrac{1}{d_o} + \dfrac{1}{d_i}

. As

d_o \to \infty

, what does

d_i

approach?

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Related Concepts

Edge Cases Limit

Background Knowledge

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

edge cases