L'Hospital's Rule - AP Calculus AB
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What must be true of $f(x)$ and $g(x)$ in L'Hospital's Rule?
What must be true of $f(x)$ and $g(x)$ in L'Hospital's Rule?
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$f(x)$ and $g(x)$ must be differentiable near $c$. Functions must be differentiable for L'Hospital's Rule to work.
$f(x)$ and $g(x)$ must be differentiable near $c$. Functions must be differentiable for L'Hospital's Rule to work.
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Find the limit: $\lim_{x \to 0} \frac{\tan x}{x}$ using L'Hospital's Rule.
Find the limit: $\lim_{x \to 0} \frac{\tan x}{x}$ using L'Hospital's Rule.
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- Derivative of $\tan x$ is $\sec^2 x$; $\frac{\sec^2 0}{1} = 1$.
- Derivative of $\tan x$ is $\sec^2 x$; $\frac{\sec^2 0}{1} = 1$.
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Define indeterminate form.
Define indeterminate form.
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A form like $\frac{0}{0}$ or $\frac{\infty}{\infty}$ where limits are not immediately obvious. Forms where direct evaluation is unclear or undefined.
A form like $\frac{0}{0}$ or $\frac{\infty}{\infty}$ where limits are not immediately obvious. Forms where direct evaluation is unclear or undefined.
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Identify the indeterminate form: $\lim_{x \to 0} \frac{1 - \cos x}{x^2}$.
Identify the indeterminate form: $\lim_{x \to 0} \frac{1 - \cos x}{x^2}$.
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$\frac{0}{0}$. Both $1 - \cos 0 = 0$ and $0^2 = 0$ give $\frac{0}{0}$.
$\frac{0}{0}$. Both $1 - \cos 0 = 0$ and $0^2 = 0$ give $\frac{0}{0}$.
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What must the original limit be for L'Hospital's Rule to apply?
What must the original limit be for L'Hospital's Rule to apply?
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An indeterminate form. L'Hospital's Rule requires indeterminate forms to be valid.
An indeterminate form. L'Hospital's Rule requires indeterminate forms to be valid.
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Find the derivative of $\tan x$ for L'Hospital's Rule.
Find the derivative of $\tan x$ for L'Hospital's Rule.
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$\sec^2 x$. Derivative of tangent function.
$\sec^2 x$. Derivative of tangent function.
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List a requirement for L'Hospital's Rule to be valid.
List a requirement for L'Hospital's Rule to be valid.
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Both $f(x)$ and $g(x)$ must approach 0 or $\infty$ as $x \to c$. Essential condition for applying the rule correctly.
Both $f(x)$ and $g(x)$ must approach 0 or $\infty$ as $x \to c$. Essential condition for applying the rule correctly.
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Identify the indeterminate form: $\lim_{x \to \infty} \frac{x^2}{x^2 + x}$.
Identify the indeterminate form: $\lim_{x \to \infty} \frac{x^2}{x^2 + x}$.
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$\frac{\infty}{\infty}$. Both numerator and denominator approach infinity.
$\frac{\infty}{\infty}$. Both numerator and denominator approach infinity.
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Find the derivative of $e^x$ needed for L'Hospital's Rule.
Find the derivative of $e^x$ needed for L'Hospital's Rule.
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$e^x$. Basic exponential derivative.
$e^x$. Basic exponential derivative.
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Find the limit: $\lim_{x \to 0} \frac{x^2}{\sin x}$ using L'Hospital's Rule.
Find the limit: $\lim_{x \to 0} \frac{x^2}{\sin x}$ using L'Hospital's Rule.
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- Apply L'Hospital's Rule: $\lim_{x \to 0} \frac{2x}{\cos x} = 0$.
- Apply L'Hospital's Rule: $\lim_{x \to 0} \frac{2x}{\cos x} = 0$.
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Find the limit using L'Hospital's Rule: $\lim_{x \to 0} \frac{e^x - 1}{x}$.
Find the limit using L'Hospital's Rule: $\lim_{x \to 0} \frac{e^x - 1}{x}$.
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- Derivative of $e^x - 1$ is $e^x$; derivative of $x$ is $1$; $\frac{e^0}{1} = 1$.
- Derivative of $e^x - 1$ is $e^x$; derivative of $x$ is $1$; $\frac{e^0}{1} = 1$.
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What is the derivative of $\sin x$ needed for L'Hospital's Rule?
What is the derivative of $\sin x$ needed for L'Hospital's Rule?
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$\cos x$. Basic trigonometric derivative.
$\cos x$. Basic trigonometric derivative.
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Find the limit: $\lim_{x \to \infty} \frac{x^2}{e^x}$ using L'Hospital's Rule.
Find the limit: $\lim_{x \to \infty} \frac{x^2}{e^x}$ using L'Hospital's Rule.
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- Apply L'Hospital's Rule repeatedly; exponential grows faster than polynomial.
- Apply L'Hospital's Rule repeatedly; exponential grows faster than polynomial.
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State L'Hospital's Rule for $\frac{\infty}{\infty}$ form.
State L'Hospital's Rule for $\frac{\infty}{\infty}$ form.
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$\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$ if $\frac{\infty}{\infty}$ form. Same rule applies for $\frac{\infty}{\infty}$ form.
$\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$ if $\frac{\infty}{\infty}$ form. Same rule applies for $\frac{\infty}{\infty}$ form.
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What is L'Hospital's Rule used for?
What is L'Hospital's Rule used for?
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Determining limits of indeterminate forms like $\frac{0}{0}$ or $\frac{\infty}{\infty}$. Applies when direct substitution gives undefined ratios.
Determining limits of indeterminate forms like $\frac{0}{0}$ or $\frac{\infty}{\infty}$. Applies when direct substitution gives undefined ratios.
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State L'Hospital's Rule for $\frac{0}{0}$ form.
State L'Hospital's Rule for $\frac{0}{0}$ form.
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$\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$ if $\frac{0}{0}$ form. Take derivatives of numerator and denominator separately.
$\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$ if $\frac{0}{0}$ form. Take derivatives of numerator and denominator separately.
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Identify the indeterminate form: $\lim_{x \to 0} \frac{\sin x}{x}$.
Identify the indeterminate form: $\lim_{x \to 0} \frac{\sin x}{x}$.
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$\frac{0}{0}$. Both $\sin 0 = 0$ and $0$ in denominator give $\frac{0}{0}$.
$\frac{0}{0}$. Both $\sin 0 = 0$ and $0$ in denominator give $\frac{0}{0}$.
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Find the limit: $\lim_{x \to 0} \frac{\ln(1+x)}{x}$ using L'Hospital's Rule.
Find the limit: $\lim_{x \to 0} \frac{\ln(1+x)}{x}$ using L'Hospital's Rule.
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- Derivative of $\ln(1+x)$ is $\frac{1}{1+x}$; $\frac{\frac{1}{1+0}}{1} = 1$.
- Derivative of $\ln(1+x)$ is $\frac{1}{1+x}$; $\frac{\frac{1}{1+0}}{1} = 1$.
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Identify the indeterminate form: $\lim_{x \to \infty} \frac{x}{e^x}$.
Identify the indeterminate form: $\lim_{x \to \infty} \frac{x}{e^x}$.
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$\frac{\infty}{\infty}$. Both $x \to \infty$ and $e^x \to \infty$ give $\frac{\infty}{\infty}$.
$\frac{\infty}{\infty}$. Both $x \to \infty$ and $e^x \to \infty$ give $\frac{\infty}{\infty}$.
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Find the derivative of $\ln x$ for L'Hospital's Rule.
Find the derivative of $\ln x$ for L'Hospital's Rule.
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$\frac{1}{x}$. Basic logarithmic derivative.
$\frac{1}{x}$. Basic logarithmic derivative.
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Find the limit: $\lim_{x \to 1} \frac{x^3 - 1}{x - 1}$ using L'Hospital's Rule.
Find the limit: $\lim_{x \to 1} \frac{x^3 - 1}{x - 1}$ using L'Hospital's Rule.
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- Derivative of $x^3 - 1$ is $3x^2$; derivative of $x - 1$ is $1$; $\frac{3(1)^2}{1} = 3$.
- Derivative of $x^3 - 1$ is $3x^2$; derivative of $x - 1$ is $1$; $\frac{3(1)^2}{1} = 3$.
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Find the limit: $\lim_{x \to \infty} \frac{\ln(x^2)}{x}$ using L'Hospital's Rule.
Find the limit: $\lim_{x \to \infty} \frac{\ln(x^2)}{x}$ using L'Hospital's Rule.
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- Use chain rule: derivative of $\ln(x^2)$ is $\frac{2}{x}$; limit is $0$.
- Use chain rule: derivative of $\ln(x^2)$ is $\frac{2}{x}$; limit is $0$.
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Find the derivative of $\sec x$ for L'Hospital's Rule.
Find the derivative of $\sec x$ for L'Hospital's Rule.
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$\sec x \tan x$. Derivative of secant function.
$\sec x \tan x$. Derivative of secant function.
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Find the limit: $\lim_{x \to 0} \frac{1 - \cos x}{x}$ using L'Hospital's Rule.
Find the limit: $\lim_{x \to 0} \frac{1 - \cos x}{x}$ using L'Hospital's Rule.
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- Derivative of $1 - \cos x$ is $\sin x$; $\frac{\sin 0}{1} = 0$.
- Derivative of $1 - \cos x$ is $\sin x$; $\frac{\sin 0}{1} = 0$.
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Does L'Hospital's Rule apply to $\lim_{x \to 0} \frac{1}{x}$?
Does L'Hospital's Rule apply to $\lim_{x \to 0} \frac{1}{x}$?
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No, it's not an indeterminate form. The limit approaches $\pm\infty$, not an indeterminate form.
No, it's not an indeterminate form. The limit approaches $\pm\infty$, not an indeterminate form.
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Find the derivative of $\cos x$ for L'Hospital's Rule.
Find the derivative of $\cos x$ for L'Hospital's Rule.
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$-\sin x$. Basic trigonometric derivative.
$-\sin x$. Basic trigonometric derivative.
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Find the limit: $\lim_{x \to \infty} \frac{2x + 3}{x + 1}$ using L'Hospital's Rule.
Find the limit: $\lim_{x \to \infty} \frac{2x + 3}{x + 1}$ using L'Hospital's Rule.
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- Apply L'Hospital's Rule: $\lim_{x \to \infty} \frac{2}{1} = 2$.
- Apply L'Hospital's Rule: $\lim_{x \to \infty} \frac{2}{1} = 2$.
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Identify the indeterminate form: $\lim_{x \to 1} \frac{x^2 - 1}{x - 1}$.
Identify the indeterminate form: $\lim_{x \to 1} \frac{x^2 - 1}{x - 1}$.
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$\frac{0}{0}$. Factoring gives $(x+1)(x-1)$ in numerator; creates $\frac{0}{0}$.
$\frac{0}{0}$. Factoring gives $(x+1)(x-1)$ in numerator; creates $\frac{0}{0}$.
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Find the derivative of $\ln(1+x)$ for L'Hospital's Rule.
Find the derivative of $\ln(1+x)$ for L'Hospital's Rule.
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$\frac{1}{1+x}$. Chain rule derivative of logarithmic function.
$\frac{1}{1+x}$. Chain rule derivative of logarithmic function.
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Find the limit: $\lim_{x \to \infty} \frac{x}{x^2 + 1}$ using L'Hospital's Rule.
Find the limit: $\lim_{x \to \infty} \frac{x}{x^2 + 1}$ using L'Hospital's Rule.
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- Apply L'Hospital's Rule: $\lim_{x \to \infty} \frac{1}{2x} = 0$.
- Apply L'Hospital's Rule: $\lim_{x \to \infty} \frac{1}{2x} = 0$.
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