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Limits (Standard)

Multi-step exam-style items.

Phase 2 · AP Calculus

FoundationsStandardAdvanced

Limits: core idea

lim⁡x→0sin⁡kxx=k\lim_{x\to0}\frac{\sin kx}{x}=k.

Example problem: hint and answer

Evaluate lim⁡x→0ln⁡(1+x)x\lim_{x\to0}\dfrac{\ln(1+x)}{x}.

Hint: Derivative of ln⁡\ln at 1. Answer: 1.

Practice (9 problems — type your answers)

  • 1. Evaluate lim⁡x→0sin⁡(3x)sin⁡(5x)\lim_{x\to0}\dfrac{\sin(3x)}{\sin(5x)}.

  • 2. Evaluate lim⁡x→∞(1+1x)x\lim_{x\to\infty}\left(1+\dfrac1x\right)^x.

  • 3. Evaluate lim⁡x→01+x−1x\lim_{x\to0}\dfrac{\sqrt{1+x}-1}{x}.

  • 4. Evaluate lim⁡x→0e2x−1sin⁡x\lim_{x\to0}\dfrac{e^{2x}-1}{\sin x}.

  • 5. Evaluate lim⁡x→0cos⁡x−1x\lim_{x\to0}\dfrac{\cos x-1}{x}.

  • 6. Evaluate lim⁡x→∞(x2+4x−x)\lim_{x\to\infty}\left(\sqrt{x^2+4x}-x\right).

  • 7. Evaluate lim⁡x→0+xln⁡x\lim_{x\to0^+}x\ln x.

  • 8. Evaluate lim⁡x→0x−sin⁡xx3\lim_{x\to0}\dfrac{x-\sin x}{x^3}.

  • 9. Evaluate lim⁡x→1x3−1x−1\lim_{x\to1}\dfrac{x^3-1}{x-1}.

Limits Practice Problems — Standard