Determining Limits Using Algebraic Manipulation - AP Calculus BC

1. What is $\lim_{x\to 1}\frac{x-1}{\sqrt{x}-1}$?

   $2$. Factor numerator: $\frac{(\sqrt{x}-1)(\sqrt{x}+1)}{\sqrt{x}-1} = \sqrt{x}+1$.

2. What is $\lim_{x\to 0}\frac{\frac{1}{x+1}-1}{x}$?

   $-1$. Simplify to $\frac{-x}{x(x+1)}=\frac{-1}{x+1}$, then substitute $x=0$.

3. What is $\lim_{x\to 0}\frac{x}{\sqrt{1+x}-1}$ after rationalizing?

   $2$. After rationalizing, get $\frac{x(\sqrt{1+x}+1)}{x}=\sqrt{1+x}+1$, substitute $x=0$.

4. What is $\lim_{x\to 2}\frac{x^2-4}{x^2-3x+2}$?

   $4$. Factor: $\frac{(x-2)(x+2)}{(x-2)(x-1)} = \frac{x+2}{x-1}$, then substitute.

5. What algebraic technique is most appropriate when $\lim_{x\to a}\frac{f(x)}{g(x)}$ gives $\frac{0}{0}$ and $f,g$ are polynomials?

   Factor numerator and denominator, then cancel the common factor. This eliminates the indeterminate form $\frac{0}{0}$.

6. What is the standard limit value $\lim_{u\to 0}\frac{1-\cos u}{u^2}$ used in algebraic manipulation?

   $\frac{1}{2}$. Derived from $\sin^2u=\frac{1-\cos(2u)}{2}$ and double angle formula.

7. What is $\lim_{x\to 1}\frac{\frac{1}{x}-1}{x-1}$?

   $-1$. Rewrite as $\frac{1-x}{x(x-1)} = \frac{-(x-1)}{x(x-1)} = \frac{-1}{x}$.

8. What is $\lim_{x\to 0}\frac{(x+1)^5-1}{x}$?

   $5$. This is the derivative of $(x+1)^5$ at $x=0$ using the difference quotient.

9. What is $\lim_{x\to 0}\frac{1-\cos(3x)}{x^2}$ using standard trig limit algebra?

   $\frac{9}{2}$. Use $\frac{1-\cos(3x)}{x^2}=9\cdot\frac{1-\cos(3x)}{(3x)^2}$ with standard limit $\frac{1}{2}$.

10. What is $\lim_{x\to 0}\frac{\sqrt{9+x}-3}{x}$?

    $\frac{1}{6}$. Rationalize to get $\frac{x}{x(\sqrt{9+x}+3)} = \frac{1}{\sqrt{9+x}+3}$.

11. What is $\lim_{x\to -2}\frac{x^2+5x+6}{x+2}$?

    $1$. Factor: $\frac{(x+2)(x+3)}{x+2} = x+3$, then substitute $x=-2$.

12. What is $\lim_{x\to 3}\frac{x^2-9}{x-3}$?

    $6$. Factor: $\frac{(x-3)(x+3)}{x-3} = x+3$, then substitute $x=3$.

13. What is $\lim_{h\to 0}\frac{(a+h)^2-a^2}{h}$?

    $2a$. Expand: $\frac{a^2+2ah+h^2-a^2}{h} = \frac{2ah+h^2}{h} = 2a+h$.

14. What is $\lim_{h\to 0}\frac{(a+h)^3-a^3}{h}$?

    $3a^2$. Expand and simplify: $\frac{3a^2h+3ah^2+h^3}{h} = 3a^2+3ah+h^2$.

15. What is $\lim_{x\to 3}\frac{x^2-9}{x-3}$?

    $6$. Factor as $\frac{(x-3)(x+3)}{x-3}$, cancel to get $x+3$, then substitute $x=3$.

16. What is $\lim_{x\to 2}\frac{x^2-4}{x-2}$?

    $4$. Factor as $\frac{(x-2)(x+2)}{x-2}$, cancel to get $x+2$, then substitute $x=2$.

17. What algebraic technique is most appropriate for $\lim_{x\to a}\frac{\sqrt{u(x)}-\sqrt{v(x)}}{w(x)}$ when direct substitution gives $\frac{0}{0}$?

    Multiply by the conjugate of the numerator to rationalize. Creates a difference of squares in the numerator.

18. What identity is used to factor $a^3-b^3$ for limit simplification?

    $a^3-b^3=(a-b)(a^2+ab+b^2)$. This is the difference of cubes factorization formula.

19. What is $\lim_{x\to 2}\frac{x^3-8}{x-2}$?

    $12$. Factor as $\frac{(x-2)(x^2+2x+4)}{x-2}$, cancel, then substitute to get $4+4+4$.

20. What is $\lim_{x\to 0}\frac{\sqrt{1+x}-1}{x}$?

    $\frac{1}{2}$. Multiply by conjugate $\frac{\sqrt{1+x}+1}{\sqrt{1+x}+1}$ to get $\frac{1}{\sqrt{1+x}+1}$.

21. What is $\lim_{x\to 2}\frac{x^3-8}{x-2}$?

    $12$. Use $x^3-8=(x-2)(x^2+2x+4)$, cancel, then substitute $x=2$.

22. What is $\lim_{x\to 0}\frac{(1+x)^5-1}{x}$?

    $5$. This is the derivative of $(1+x)^5$ at $x=0$ using difference quotient.

23. What is $\lim_{x\to 0}\frac{\sqrt{x+9}-3}{x}$?

    $\frac{1}{6}$. Multiply by conjugate $\frac{\sqrt{x+9}+3}{\sqrt{x+9}+3}$ to get $\frac{1}{\sqrt{x+9}+3}$.

24. What is $\lim_{x\to 1}\frac{x^3-1}{x-1}$?

    $3$. Factor as $\frac{(x-1)(x^2+x+1)}{x-1}$, cancel, then substitute to get $1+1+1$.

25. What is $\lim_{x\to 0}\frac{\tan(4x)}{x}$ using $\tan u=\frac{\sin u}{\cos u}$?

    $4$. Rewrite as $\frac{\sin(4x)}{x\cos(4x)}=4\cdot\frac{\sin(4x)}{4x}\cdot\frac{1}{\cos(4x)}$.

26. What is the standard limit value $\lim_{u\to 0}\frac{\sin u}{u}$ used for algebraic substitution?

    $1$. Fundamental trigonometric limit proven using squeeze theorem.

27. What is $\lim_{x\to -1}\frac{x^3+1}{x+1}$?

    $3$. Use $x^3+1=(x+1)(x^2-x+1)$, cancel, then substitute $x=-1$.

28. What is $\lim_{x\to 0}\frac{\sqrt{1+x}-1}{x}$?

    $\frac{1}{2}$. Multiply by conjugate: $\frac{x}{x(\sqrt{1+x}+1)} = \frac{1}{\sqrt{1+x}+1}$.

29. What is $\lim_{x\to 1}\frac{x^3-1}{x-1}$?

    $3$. Use $x^3-1=(x-1)(x^2+x+1)$, cancel, then substitute $x=1$.

30. What is $\lim_{x\to -1}\frac{x^2+3x+2}{x+1}$?

    $1$. Factor as $\frac{(x+1)(x+2)}{x+1}$, cancel to get $x+2$, then substitute $x=-1$.

31. What algebraic technique is most common when $\lim_{x\to a}\frac{f(x)}{g(x)}$ gives $\frac{0}{0}$ and polynomials are involved?

    Factor numerator and denominator, then cancel the common factor. Factoring reveals removable discontinuities when direct substitution yields $\frac{0}{0}$.

32. What is $\lim_{x\to 0}\frac{\sin(5x)}{x}$ using standard trig limit algebra?

    $5$. Rewrite as $5\cdot\frac{\sin(5x)}{5x}$ and use $\lim_{u\to 0}\frac{\sin u}{u}=1$.

33. What identity is used to factor $a^3+b^3$ for limit simplification?

    $a^3+b^3=(a+b)(a^2-ab+b^2)$. This is the sum of cubes factorization formula.

34. What is the conjugate of $a+b$ that is used to simplify radicals in limits?

    $a-b$. Multiplying by the conjugate eliminates square roots in the denominator.

35. What is $\lim_{x\to 1}\frac{x^2-1}{x^3-1}$?

    $\frac{2}{3}$. Factor as $\frac{(x-1)(x+1)}{(x-1)(x^2+x+1)}$, cancel and substitute.

36. What is $\lim_{x\to 0}\frac{\frac{1}{x+1}-1}{x}$?

    $-1$. Simplify: $\frac{\frac{1-(x+1)}{x+1}}{x} = \frac{-x}{x(x+1)} = \frac{-1}{x+1}$.

37. What is $\lim_{x\to 0}\frac{(1+2x)^3-1}{x}$?

    $6$. Derivative of $(1+2x)^3$ at $x=0$: $3(1+2x)^2 \cdot 2 = 6$.

38. What is $\lim_{x\to 4}\frac{\sqrt{x}-2}{x-4}$?

    $\frac{1}{4}$. Multiply by conjugate to get $\frac{1}{\sqrt{x}+2}$, then substitute $x=4$.

39. What is $\lim_{x\to 4}\frac{\sqrt{x}-2}{x-4}$?

    $\frac{1}{4}$. Multiply by $\frac{\sqrt{x}+2}{\sqrt{x}+2}$ to get $\frac{1}{\sqrt{x}+2}$.

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