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

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In the ideal gas law PV=nRTPV = nRT, what does pressure PP approach as volume Vโ†’โˆžV\to\infty at fixed T,nT, n?

Example 2

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

Example 3

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

Example 4

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

Example 5

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What does 1n2\dfrac{1}{n^2} approach as nโ†’โˆžn\to\infty?

Example 6

easy
As xโ†’0+x \to 0^+, what does lnโกx\ln x approach?

Example 7

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Check the formula for cylinder volume V=ฯ€r2hV=\pi r^2 h in the limit hโ†’0h\to 0. What should VV approach?

Example 8

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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

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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

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Population grows as P(t)=P0ektP(t) = P_0 e^{kt} with k>0k>0. What does PP approach as tโ†’โˆžt\to\infty?

Example 12

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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 nn large and p=ฮป/np = \lambda/n, find the limit of P(X=k)=(nk)pk(1โˆ’p)nโˆ’kP(X=k) = \binom{n}{k}p^k(1-p)^{n-k}.

Example 14

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

Example 15

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As nโ†’โˆžn \to \infty, what does 2n+1n+3\dfrac{2n+1}{n+3} approach?

Example 16

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As nโ†’โˆžn\to\infty, what does nn\sqrt[n]{n} approach?

Example 17

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For the partial sum Sn=1+1/2+1/4+โ€ฆ+1/2nโˆ’1S_n = 1 + 1/2 + 1/4 + \ldots + 1/2^{n-1}, what does SnS_n approach as nโ†’โˆžn\to\infty?

Example 18

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

Example 19

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A relativistic energy formula gives E=mc2/1โˆ’v2/c2E = mc^2/\sqrt{1 - v^2/c^2}. What does EE approach as vโ†’0v\to 0?

Example 20

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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 CasesLimit