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Unit 1: Limits and Continuity

Study guide for CLEP CLEP Calculus — Unit 1: Limits and Continuity. Practice questions, key concepts, and exam tips.

284

Practice Questions

30

Flashcards

4

Key Topics

Key Concepts to Study

limit evaluation techniques
one-sided limits
continuity and IVT
limits at infinity

Sample Practice Questions

Try these 5 questions from this unit. Sign up for full access to all 284.

Q1MEDIUM

If a function f(x) is differentiable at x = a, what can be concluded about its continuity at x = a?

A) It is not continuous
B) It is continuous
C) It is not defined
D) The limit does not exist
E) The derivative is zero
Show Answer

Answer: B — It is continuous is correct because differentiability implies continuity, since the existence of a derivative at a point requires the function to have a well-defined limit at that point, which is the definition of continuity.

Q2MEDIUM

Find the slope of y = x^3 - 2x^2 + x - 1 at x = 3.

A) 10
B) 12
C) 14
D) 16
E) 20
Show Answer

Answer: D — 16 is correct because the derivative of y = x^3 - 2x^2 + x - 1 is y' = 3x^2 - 4x + 1, and at x = 3, y' = 3*3^2 - 4*3 + 1 = 27 - 12 + 1 = 16.

Q3MEDIUM

Find the derivative of y = $tan^{2}$(x).

A) 2tan(x)$sec^{2}$(x)
B) 2tan(x)csc(x)
C) 2sec(x)tan(x)
D) 2sec^2(x)
E) 2csc(x)cot(x)
Show Answer

Answer: A — Write y = (tan x)^2 and apply the chain rule: dy/dx = 2·tan(x)·(derivative of tan x) = 2·tan(x)·sec^2(x), since the derivative of tan(x) is sec^2(x). The choice 2·sec(x)·tan(x) is the derivative of sec(x), not of tan^2(x), so it is the common trap here.

Q4MEDIUM

Find the limit of the function f(x) = (e^x - 1) / x as x approaches 0 using L'Hospital's Rule.

A) The limit is infinity
B) The limit is 0
C) The limit is undefined
D) The limit is 1
E) The limit is -1
Show Answer

Answer: D — Apply L'Hospital's Rule to find the limit. The derivative of the numerator is e^x, and the derivative of the denominator is 1. So, the limit is $e^{0}$ / 1 = 1.

Q5EASY

What is the limit as x approaches infinity of e^(-x)?

A) e^2
B) 0
C) 1
D) -1
E) e
Show Answer

Answer: B — 0 is correct because as x approaches infinity, e^(-x) approaches 0 since the negative exponent causes the function value to decrease towards 0.

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Study Tips for Unit 1: Limits and Continuity

  • Focus on understanding concepts, not memorizing facts — CLEP tests application
  • Practice with timed questions to build exam-day speed
  • Review explanations for wrong answers — they reveal common misconceptions
  • Use flashcards for key terms, practice questions for deeper understanding

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