Determining Limits Using Algebraic Manipulation - AP Calculus BC
Card 1 of 30
What is $\lim_{x\to 3}\frac{x^2-9}{x-3}$?
What is $\lim_{x\to 3}\frac{x^2-9}{x-3}$?
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$6$. Factor as $\frac{(x-3)(x+3)}{x-3}$, cancel to get $x+3$, then substitute $x=3$.
$6$. Factor as $\frac{(x-3)(x+3)}{x-3}$, cancel to get $x+3$, then substitute $x=3$.
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What is $\lim_{x\to 0}\frac{\sin(5x)}{x}$ using standard trig limit algebra?
What is $\lim_{x\to 0}\frac{\sin(5x)}{x}$ using standard trig limit algebra?
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$5$. Rewrite as $5\cdot\frac{\sin(5x)}{5x}$ and use $\lim_{u\to 0}\frac{\sin u}{u}=1$.
$5$. Rewrite as $5\cdot\frac{\sin(5x)}{5x}$ and use $\lim_{u\to 0}\frac{\sin u}{u}=1$.
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What is $\lim_{x\to 0}\frac{\frac{1}{x+1}-1}{x}$?
What is $\lim_{x\to 0}\frac{\frac{1}{x+1}-1}{x}$?
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$-1$. Simplify to $\frac{-x}{x(x+1)}=\frac{-1}{x+1}$, then substitute $x=0$.
$-1$. Simplify to $\frac{-x}{x(x+1)}=\frac{-1}{x+1}$, then substitute $x=0$.
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What is $\lim_{x\to 0}\frac{(x+1)^5-1}{x}$?
What is $\lim_{x\to 0}\frac{(x+1)^5-1}{x}$?
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$5$. This is the derivative of $(x+1)^5$ at $x=0$ using the difference quotient.
$5$. This is the derivative of $(x+1)^5$ at $x=0$ using the difference quotient.
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What is $\lim_{x\to 4}\frac{\sqrt{x}-2}{x-4}$?
What is $\lim_{x\to 4}\frac{\sqrt{x}-2}{x-4}$?
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$\frac{1}{4}$. Multiply by conjugate to get $\frac{1}{\sqrt{x}+2}$, then substitute $x=4$.
$\frac{1}{4}$. Multiply by conjugate to get $\frac{1}{\sqrt{x}+2}$, then substitute $x=4$.
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What is $\lim_{x\to 0}\frac{\sqrt{1+x}-1}{x}$?
What is $\lim_{x\to 0}\frac{\sqrt{1+x}-1}{x}$?
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$\frac{1}{2}$. Multiply by conjugate $\frac{\sqrt{1+x}+1}{\sqrt{1+x}+1}$ to get $\frac{1}{\sqrt{1+x}+1}$.
$\frac{1}{2}$. Multiply by conjugate $\frac{\sqrt{1+x}+1}{\sqrt{1+x}+1}$ to get $\frac{1}{\sqrt{1+x}+1}$.
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What is $\lim_{x\to 0}\frac{\sqrt{x+9}-3}{x}$?
What is $\lim_{x\to 0}\frac{\sqrt{x+9}-3}{x}$?
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$\frac{1}{6}$. Multiply by conjugate $\frac{\sqrt{x+9}+3}{\sqrt{x+9}+3}$ to get $\frac{1}{\sqrt{x+9}+3}$.
$\frac{1}{6}$. Multiply by conjugate $\frac{\sqrt{x+9}+3}{\sqrt{x+9}+3}$ to get $\frac{1}{\sqrt{x+9}+3}$.
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What is $\lim_{x\to 2}\frac{x^3-8}{x-2}$?
What is $\lim_{x\to 2}\frac{x^3-8}{x-2}$?
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$12$. Factor as $\frac{(x-2)(x^2+2x+4)}{x-2}$, cancel, then substitute to get $4+4+4$.
$12$. Factor as $\frac{(x-2)(x^2+2x+4)}{x-2}$, cancel, then substitute to get $4+4+4$.
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What is $\lim_{x\to 1}\frac{x^3-1}{x-1}$?
What is $\lim_{x\to 1}\frac{x^3-1}{x-1}$?
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$3$. Factor as $\frac{(x-1)(x^2+x+1)}{x-1}$, cancel, then substitute to get $1+1+1$.
$3$. Factor as $\frac{(x-1)(x^2+x+1)}{x-1}$, cancel, then substitute to get $1+1+1$.
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What is $\lim_{x\to -1}\frac{x^2+3x+2}{x+1}$?
What is $\lim_{x\to -1}\frac{x^2+3x+2}{x+1}$?
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$1$. Factor as $\frac{(x+1)(x+2)}{x+1}$, cancel to get $x+2$, then substitute $x=-1$.
$1$. Factor as $\frac{(x+1)(x+2)}{x+1}$, cancel to get $x+2$, then substitute $x=-1$.
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What is $\lim_{x\to 2}\frac{x^2-4}{x-2}$?
What is $\lim_{x\to 2}\frac{x^2-4}{x-2}$?
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$4$. Factor as $\frac{(x-2)(x+2)}{x-2}$, cancel to get $x+2$, then substitute $x=2$.
$4$. Factor as $\frac{(x-2)(x+2)}{x-2}$, cancel to get $x+2$, then substitute $x=2$.
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What is the conjugate of $a+b$ that is used to simplify radicals in limits?
What is the conjugate of $a+b$ that is used to simplify radicals in limits?
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$a-b$. Multiplying by the conjugate eliminates square roots in the denominator.
$a-b$. Multiplying by the conjugate eliminates square roots in the denominator.
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What is $\lim_{x\to 0}\frac{\tan(4x)}{x}$ using $\tan u=\frac{\sin u}{\cos u}$?
What is $\lim_{x\to 0}\frac{\tan(4x)}{x}$ using $\tan u=\frac{\sin u}{\cos u}$?
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$4$. Rewrite as $\frac{\sin(4x)}{x\cos(4x)}=4\cdot\frac{\sin(4x)}{4x}\cdot\frac{1}{\cos(4x)}$.
$4$. Rewrite as $\frac{\sin(4x)}{x\cos(4x)}=4\cdot\frac{\sin(4x)}{4x}\cdot\frac{1}{\cos(4x)}$.
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What is the standard limit value $\lim_{u\to 0}\frac{1-\cos u}{u^2}$ used in algebraic manipulation?
What is the standard limit value $\lim_{u\to 0}\frac{1-\cos u}{u^2}$ used in algebraic manipulation?
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$\frac{1}{2}$. Derived from $\sin^2u=\frac{1-\cos(2u)}{2}$ and double angle formula.
$\frac{1}{2}$. Derived from $\sin^2u=\frac{1-\cos(2u)}{2}$ and double angle formula.
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What is the standard limit value $\lim_{u\to 0}\frac{\sin u}{u}$ used for algebraic substitution?
What is the standard limit value $\lim_{u\to 0}\frac{\sin u}{u}$ used for algebraic substitution?
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$1$. Fundamental trigonometric limit proven using squeeze theorem.
$1$. Fundamental trigonometric limit proven using squeeze theorem.
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What algebraic technique is most common when $\lim_{x\to a}\frac{f(x)}{g(x)}$ gives $\frac{0}{0}$ and polynomials are involved?
What algebraic technique is most common when $\lim_{x\to a}\frac{f(x)}{g(x)}$ gives $\frac{0}{0}$ and polynomials are involved?
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Factor numerator and denominator, then cancel the common factor. Factoring reveals removable discontinuities when direct substitution yields $\frac{0}{0}$.
Factor numerator and denominator, then cancel the common factor. Factoring reveals removable discontinuities when direct substitution yields $\frac{0}{0}$.
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What is $\lim_{x\to 0}\frac{1-\cos(3x)}{x^2}$ using standard trig limit algebra?
What is $\lim_{x\to 0}\frac{1-\cos(3x)}{x^2}$ using standard trig limit algebra?
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$\frac{9}{2}$. Use $\frac{1-\cos(3x)}{x^2}=9\cdot\frac{1-\cos(3x)}{(3x)^2}$ with standard limit $\frac{1}{2}$.
$\frac{9}{2}$. Use $\frac{1-\cos(3x)}{x^2}=9\cdot\frac{1-\cos(3x)}{(3x)^2}$ with standard limit $\frac{1}{2}$.
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What is $\lim_{x\to 1}\frac{x^2-1}{x^3-1}$?
What is $\lim_{x\to 1}\frac{x^2-1}{x^3-1}$?
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$\frac{2}{3}$. Factor as $\frac{(x-1)(x+1)}{(x-1)(x^2+x+1)}$, cancel and substitute.
$\frac{2}{3}$. Factor as $\frac{(x-1)(x+1)}{(x-1)(x^2+x+1)}$, cancel and substitute.
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What is $\lim_{x\to 0}\frac{x}{\sqrt{1+x}-1}$ after rationalizing?
What is $\lim_{x\to 0}\frac{x}{\sqrt{1+x}-1}$ after rationalizing?
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$2$. After rationalizing, get $\frac{x(\sqrt{1+x}+1)}{x}=\sqrt{1+x}+1$, substitute $x=0$.
$2$. After rationalizing, get $\frac{x(\sqrt{1+x}+1)}{x}=\sqrt{1+x}+1$, substitute $x=0$.
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What is $ \lim_{x\to 2}\frac{x^3-8}{x-2}$?
What is $ \lim_{x\to 2}\frac{x^3-8}{x-2}$?
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$12$. Use $x^3-8=(x-2)(x^2+2x+4)$, cancel, then substitute $x=2$.
$12$. Use $x^3-8=(x-2)(x^2+2x+4)$, cancel, then substitute $x=2$.
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What identity is used to factor $a^3-b^3$ for limit simplification?
What identity is used to factor $a^3-b^3$ for limit simplification?
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$a^3-b^3=(a-b)(a^2+ab+b^2)$. This is the difference of cubes factorization formula.
$a^3-b^3=(a-b)(a^2+ab+b^2)$. This is the difference of cubes factorization formula.
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What is $ \lim_{x\to -1}\frac{x^3+1}{x+1}$?
What is $ \lim_{x\to -1}\frac{x^3+1}{x+1}$?
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$3$. Use $x^3+1=(x+1)(x^2-x+1)$, cancel, then substitute $x=-1$.
$3$. Use $x^3+1=(x+1)(x^2-x+1)$, cancel, then substitute $x=-1$.
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What is $ \lim_{h\to 0}\frac{(a+h)^3-a^3}{h}$?
What is $ \lim_{h\to 0}\frac{(a+h)^3-a^3}{h}$?
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$3a^2$. Expand and simplify: $\frac{3a^2h+3ah^2+h^3}{h} = 3a^2+3ah+h^2$.
$3a^2$. Expand and simplify: $\frac{3a^2h+3ah^2+h^3}{h} = 3a^2+3ah+h^2$.
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What is $ \lim_{x\to 3}\frac{x^2-9}{x-3}$?
What is $ \lim_{x\to 3}\frac{x^2-9}{x-3}$?
Tap to reveal answer
$6$. Factor: $\frac{(x-3)(x+3)}{x-3} = x+3$, then substitute $x=3$.
$6$. Factor: $\frac{(x-3)(x+3)}{x-3} = x+3$, then substitute $x=3$.
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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}$?
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}$?
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Multiply by the conjugate of the numerator to rationalize. Creates a difference of squares in the numerator.
Multiply by the conjugate of the numerator to rationalize. Creates a difference of squares in the numerator.
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What is $ \lim_{x\to 4}\frac{\sqrt{x}-2}{x-4}$?
What is $ \lim_{x\to 4}\frac{\sqrt{x}-2}{x-4}$?
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$\frac{1}{4}$. Multiply by $\frac{\sqrt{x}+2}{\sqrt{x}+2}$ to get $\frac{1}{\sqrt{x}+2}$.
$\frac{1}{4}$. Multiply by $\frac{\sqrt{x}+2}{\sqrt{x}+2}$ to get $\frac{1}{\sqrt{x}+2}$.
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What is $ \lim_{x\to 2}\frac{x^2-4}{x^2-3x+2}$?
What is $ \lim_{x\to 2}\frac{x^2-4}{x^2-3x+2}$?
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$4$. Factor: $\frac{(x-2)(x+2)}{(x-2)(x-1)} = \frac{x+2}{x-1}$, then substitute.
$4$. Factor: $\frac{(x-2)(x+2)}{(x-2)(x-1)} = \frac{x+2}{x-1}$, then substitute.
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What is $ \lim_{x\to -2}\frac{x^2+5x+6}{x+2}$?
What is $ \lim_{x\to -2}\frac{x^2+5x+6}{x+2}$?
Tap to reveal answer
$1$. Factor: $\frac{(x+2)(x+3)}{x+2} = x+3$, then substitute $x=-2$.
$1$. Factor: $\frac{(x+2)(x+3)}{x+2} = x+3$, then substitute $x=-2$.
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What is $ \lim_{x\to 0}\frac{\frac{1}{x+1}-1}{x}$?
What is $ \lim_{x\to 0}\frac{\frac{1}{x+1}-1}{x}$?
Tap to reveal answer
$-1$. Simplify: $\frac{\frac{1-(x+1)}{x+1}}{x} = \frac{-x}{x(x+1)} = \frac{-1}{x+1}$.
$-1$. Simplify: $\frac{\frac{1-(x+1)}{x+1}}{x} = \frac{-x}{x(x+1)} = \frac{-1}{x+1}$.
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What is $ \lim_{x\to 1}\frac{\frac{1}{x}-1}{x-1}$?
What is $ \lim_{x\to 1}\frac{\frac{1}{x}-1}{x-1}$?
Tap to reveal answer
$-1$. Rewrite as $\frac{1-x}{x(x-1)} = \frac{-(x-1)}{x(x-1)} = \frac{-1}{x}$.
$-1$. Rewrite as $\frac{1-x}{x(x-1)} = \frac{-(x-1)}{x(x-1)} = \frac{-1}{x}$.
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