1Calculate lim x→3 (x²−9)/(x−3) and identify the excluded input.
Show a complete solution and verify with a second representation.
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.
2Rationalize and evaluate lim x→4 [√(x+5)−3]/(x−4).
Show a complete solution and verify with a second representation.
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.
3Evaluate lim x→0 (x²−4x)/x, with x≠0.
Show a complete solution and verify with a second representation.
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.
4Evaluate lim x→0 sin(5x)/x with x measured in radians.
Show a complete solution and verify with a second representation.
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.
5Evaluate lim x→0 [1−cos(x)]/x using a trigonometric identity.
Show a complete solution and verify with a second representation.
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.
6For f(x)=(x²−1)/(x−1) if x≠1 and f(1)=9, classify continuity at x=1.
Show a complete solution and verify with a second representation.
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.
7Compute lim x→−2 (x²+3x+2)/(x+2).
Show a complete solution and verify with a second representation.
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.
8Find the horizontal asymptote of f(x)=(2x²+3)/(x²−1).
Show a complete solution and verify with a second representation.
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.
9Calculate lim x→∞ (3x²−x+1)/(2x²+5).
Show a complete solution and verify with a second representation.
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.
10Evaluate lim x→∞ x/(x²+1), explaining the dominant terms.
Show a complete solution and verify with a second representation.
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.
11Create a left/right table for f(x)=x² near x=2 and state its limit.
Show a complete solution and verify with a second representation.
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.
12Evaluate lim x→0 tan(x)/x in radians.
Show a complete solution and verify with a second representation.
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.
13For f(x)=x² when x<0 and f(x)=x+1 when x≥0, evaluate lim x→0 f(x).
Show a complete solution and verify with a second representation.
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.
14Evaluate lim x→∞ [7−3/x].
Show a complete solution and verify with a second representation.
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.
15Evaluate lim x→1 (x³−1)/(x−1) for x≠1.
Show a complete solution and verify with a second representation.
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.
16For f(x)=(x²−4)/(x−2) when x≠2 and f(2)=7, compare the limit and f(2).
Show a complete solution and verify with a second representation.
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.