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Unit 02 · Extended Pathway

Limits: Numerical, Graphical & Algebraic Reasoning

How can values near an input reveal behavior at that input—even when direct substitution fails?

Khaemenes Academy · Calculus I

Extended Practice

Limits: Numerical, Graphical & Algebraic Reasoning · Show reasoning, notation, intervals, units, and corrections.

U02

Name

Date

1

Evaluate lim x→0 [√(4+x)−2]/x and justify rationalization.

Generalize, analyze an edge case, or defend the result without relying on technology.

Representation: Where appropriate, draw a labelled left/right numerical table or function sketch. State whether the approach agrees from both sides and explain any hole, jump, or asymptote.

2

Evaluate lim x→0 sin(3x)/sin(5x), in radians.

Generalize, analyze an edge case, or defend the result without relying on technology.

Representation: Where appropriate, draw a labelled left/right numerical table or function sketch. State whether the approach agrees from both sides and explain any hole, jump, or asymptote.

3

Evaluate lim x→∞ (4x³−2x)/(2x³+7) with leading-term reasoning.

Generalize, analyze an edge case, or defend the result without relying on technology.

Representation: Where appropriate, draw a labelled left/right numerical table or function sketch. State whether the approach agrees from both sides and explain any hole, jump, or asymptote.

4

Determine lim x→∞ (x²+1)/(x+2), including its sign.

Generalize, analyze an edge case, or defend the result without relying on technology.

Representation: Where appropriate, draw a labelled left/right numerical table or function sketch. State whether the approach agrees from both sides and explain any hole, jump, or asymptote.

5

Test the existence of lim x→0 |x|/x using two one-sided calculations.

Generalize, analyze an edge case, or defend the result without relying on technology.

Representation: Where appropriate, draw a labelled left/right numerical table or function sketch. State whether the approach agrees from both sides and explain any hole, jump, or asymptote.

6

Calculate lim x→2 (x³−8)/(x−2) using a difference of cubes.

Generalize, analyze an edge case, or defend the result without relying on technology.

Representation: Where appropriate, draw a labelled left/right numerical table or function sketch. State whether the approach agrees from both sides and explain any hole, jump, or asymptote.

7

Choose k so that f(x)=x²+k for x<1 and f(x)=3x for x≥1 is continuous at 1.

Generalize, analyze an edge case, or defend the result without relying on technology.

Representation: Where appropriate, draw a labelled left/right numerical table or function sketch. State whether the approach agrees from both sides and explain any hole, jump, or asymptote.

8

Evaluate lim x→0 [e^(2x)−1]/x. Identify the underlying standard limit.

Generalize, analyze an edge case, or defend the result without relying on technology.

Representation: Where appropriate, draw a labelled left/right numerical table or function sketch. State whether the approach agrees from both sides and explain any hole, jump, or asymptote.

9

Compute lim x→∞ √(x²+4)/x, carefully handling absolute value.

Generalize, analyze an edge case, or defend the result without relying on technology.

Representation: Where appropriate, draw a labelled left/right numerical table or function sketch. State whether the approach agrees from both sides and explain any hole, jump, or asymptote.

10

Compute lim x→−∞ √(x²+4)/x. Explain the different sign.

Generalize, analyze an edge case, or defend the result without relying on technology.

Representation: Where appropriate, draw a labelled left/right numerical table or function sketch. State whether the approach agrees from both sides and explain any hole, jump, or asymptote.

11

Give a function continuous at zero but not differentiable there, and justify.

Generalize, analyze an edge case, or defend the result without relying on technology.

Representation: Where appropriate, draw a labelled left/right numerical table or function sketch. State whether the approach agrees from both sides and explain any hole, jump, or asymptote.

12

Give a function that crosses its own horizontal asymptote, and prove both claims.

Generalize, analyze an edge case, or defend the result without relying on technology.

Representation: Where appropriate, draw a labelled left/right numerical table or function sketch. State whether the approach agrees from both sides and explain any hole, jump, or asymptote.

The printed questions are formative exercises. Passing course mastery checks at 80% is required to unlock the next activity; worksheet completion alone does not grant progression.

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