Practice Limiting 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

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?

Showing a random 20 of 50 problems.

Example 1

medium

In the ideal gas law

PV = nRT

, what does pressure

P

approach as volume

V\to\infty

at fixed

T, n

Example 2

medium

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

Example 3

easy

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

Example 4

medium

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

Example 5

easy

What does

\dfrac{1}{n^2}

approach as

n\to\infty

Example 6

easy

x \to 0^+

, what does

\ln x

approach?

Example 7

easy

Check the formula for cylinder volume

V=\pi r^2 h

in the limit

h\to 0

. What should

V

approach?

Example 8

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 9

easy

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

Example 10

challenge

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

Example 11

medium

Population grows as

P(t) = P_0 e^{kt}

with

k>0

. What does

P

approach as

t\to\infty

Example 12

medium

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

Example 13

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 14

easy

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

Example 15

easy

n \to \infty

, what does

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

approach?

Example 16

medium

n\to\infty

, what does

\sqrt[n]{n}

approach?

Example 17

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 18

easy

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

Example 19

hard

A relativistic energy formula gives

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

. What does

E

approach as

v\to 0

Example 20

medium

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

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Related Concepts

Edge Cases Limit