Calculus Examples

$lim_{x \to \frac{π}{4}} \frac{1 - \tan (x)}{\cot (2 x)}$

Step 1

Apply L'Hospital's rule.

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Step 1.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 1.1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{lim_{x \to \frac{π}{4}} 1 - \tan (x)}{lim_{x \to \frac{π}{4}} \cot (2 x)}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Evaluate the limit.

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Step 1.1.2.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{π}{4}$ .

$\frac{lim_{x \to \frac{π}{4}} 1 - lim_{x \to \frac{π}{4}} \tan (x)}{lim_{x \to \frac{π}{4}} \cot (2 x)}$

Step 1.1.2.1.2

Evaluate the limit of $1$ which is constant as $x$ approaches $\frac{π}{4}$ .

$\frac{1 - lim_{x \to \frac{π}{4}} \tan (x)}{lim_{x \to \frac{π}{4}} \cot (2 x)}$

Step 1.1.2.1.3

Move the limit inside the trig function because tangent is continuous.

$\frac{1 - \tan (lim_{x \to \frac{π}{4}} x)}{lim_{x \to \frac{π}{4}} \cot (2 x)}$

Step 1.1.2.2

Evaluate the limit of $x$ by plugging in $\frac{π}{4}$ for $x$ .

$\frac{1 - \tan (\frac{π}{4})}{lim_{x \to \frac{π}{4}} \cot (2 x)}$

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Simplify each term.

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Step 1.1.2.3.1.1

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$\frac{1 - 1 \cdot 1}{lim_{x \to \frac{π}{4}} \cot (2 x)}$

Step 1.1.2.3.1.2

Multiply $- 1$ by $1$ .

$\frac{1 - 1}{lim_{x \to \frac{π}{4}} \cot (2 x)}$

Step 1.1.2.3.2

Subtract $1$ from $1$ .

$\frac{0}{lim_{x \to \frac{π}{4}} \cot (2 x)}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Evaluate the limit.

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Step 1.1.3.1.1

Move the limit inside the trig function because cotangent is continuous.

$\frac{0}{\cot (lim_{x \to \frac{π}{4}} 2 x)}$

Step 1.1.3.1.2

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{\cot (2 lim_{x \to \frac{π}{4}} x)}$

Step 1.1.3.2

Evaluate the limit of $x$ by plugging in $\frac{π}{4}$ for $x$ .

$\frac{0}{\cot (2 \frac{π}{4})}$

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Cancel the common factor of $2$ .

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Step 1.1.3.3.1.1

Factor $2$ out of $4$ .

$\frac{0}{\cot (2 \frac{π}{2 (2)})}$

Step 1.1.3.3.1.2

Cancel the common factor.

$\frac{0}{\cot (2 \frac{π}{2 \cdot 2})}$

Step 1.1.3.3.1.3

Rewrite the expression.

$\frac{0}{\cot (\frac{π}{2})}$

Step 1.1.3.3.2

The exact value of $\cot (\frac{π}{2})$ is $0$ .

$\frac{0}{0}$

Step 1.1.3.3.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.1.3.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to \frac{π}{4}} \frac{1 - \tan (x)}{\cot (2 x)} = lim_{x \to \frac{π}{4}} \frac{\frac{d}{d x} [1 - \tan (x)]}{\frac{d}{d x} [\cot (2 x)]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to \frac{π}{4}} \frac{\frac{d}{d x} [1 - \tan (x)]}{\frac{d}{d x} [\cot (2 x)]}$

Step 1.3.2

By the Sum Rule, the derivative of $1 - \tan (x)$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- \tan (x)]$ .

$lim_{x \to \frac{π}{4}} \frac{\frac{d}{d x} [1] + \frac{d}{d x} [- \tan (x)]}{\frac{d}{d x} [\cot (2 x)]}$

Step 1.3.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$lim_{x \to \frac{π}{4}} \frac{0 + \frac{d}{d x} [- \tan (x)]}{\frac{d}{d x} [\cot (2 x)]}$

Step 1.3.4

Evaluate $\frac{d}{d x} [- \tan (x)]$ .

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Step 1.3.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \tan (x)$ with respect to $x$ is $- \frac{d}{d x} [\tan (x)]$ .

$lim_{x \to \frac{π}{4}} \frac{0 - \frac{d}{d x} [\tan (x)]}{\frac{d}{d x} [\cot (2 x)]}$

Step 1.3.4.2

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

$lim_{x \to \frac{π}{4}} \frac{0 - \sec^{2} (x)}{\frac{d}{d x} [\cot (2 x)]}$

Step 1.3.5

Subtract $\sec^{2} (x)$ from $0$ .

$lim_{x \to \frac{π}{4}} \frac{- \sec^{2} (x)}{\frac{d}{d x} [\cot (2 x)]}$

Step 1.3.6

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cot (x)$ and $g (x) = 2 x$ .

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Step 1.3.6.1

To apply the Chain Rule, set $u$ as $2 x$ .

$lim_{x \to \frac{π}{4}} \frac{- \sec^{2} (x)}{\frac{d}{d u} [\cot (u)] \frac{d}{d x} [2 x]}$

Step 1.3.6.2

The derivative of $\cot (u)$ with respect to $u$ is $- \csc^{2} (u)$ .

$lim_{x \to \frac{π}{4}} \frac{- \sec^{2} (x)}{- \csc^{2} (u) \frac{d}{d x} [2 x]}$

Step 1.3.6.3

Replace all occurrences of $u$ with $2 x$ .

$lim_{x \to \frac{π}{4}} \frac{- \sec^{2} (x)}{- \csc^{2} (2 x) \frac{d}{d x} [2 x]}$

Step 1.3.7

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$lim_{x \to \frac{π}{4}} \frac{- \sec^{2} (x)}{- \csc^{2} (2 x) (2 \frac{d}{d x} [x])}$

Step 1.3.8

Multiply $2$ by $- 1$ .

$lim_{x \to \frac{π}{4}} \frac{- \sec^{2} (x)}{- 2 \csc^{2} (2 x) \frac{d}{d x} [x]}$

Step 1.3.9

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to \frac{π}{4}} \frac{- \sec^{2} (x)}{- 2 \csc^{2} (2 x) \cdot 1}$

Step 1.3.10

Multiply $- 2$ by $1$ .

$lim_{x \to \frac{π}{4}} \frac{- \sec^{2} (x)}{- 2 \csc^{2} (2 x)}$

Step 2

Evaluate the limit.

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Step 2.1

Move the term $\frac{1}{- 2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{- 2} lim_{x \to \frac{π}{4}} \frac{- \sec^{2} (x)}{\csc^{2} (2 x)}$

Step 2.2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $\frac{π}{4}$ .

$\frac{1}{- 2} \cdot \frac{lim_{x \to \frac{π}{4}} - \sec^{2} (x)}{lim_{x \to \frac{π}{4}} \csc^{2} (2 x)}$

Step 2.3

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{- 2} \cdot \frac{- lim_{x \to \frac{π}{4}} \sec^{2} (x)}{lim_{x \to \frac{π}{4}} \csc^{2} (2 x)}$

Step 2.4

Move the exponent $2$ from $\sec^{2} (x)$ outside the limit using the Limits Power Rule.

$\frac{1}{- 2} \cdot \frac{- {(lim_{x \to \frac{π}{4}} \sec (x))}^{2}}{lim_{x \to \frac{π}{4}} \csc^{2} (2 x)}$

Step 2.5

Move the limit inside the trig function because secant is continuous.

$\frac{1}{- 2} \cdot \frac{- \sec^{2} (lim_{x \to \frac{π}{4}} x)}{lim_{x \to \frac{π}{4}} \csc^{2} (2 x)}$

Step 2.6

Move the exponent $2$ from $\csc^{2} (2 x)$ outside the limit using the Limits Power Rule.

$\frac{1}{- 2} \cdot \frac{- \sec^{2} (lim_{x \to \frac{π}{4}} x)}{{(lim_{x \to \frac{π}{4}} \csc (2 x))}^{2}}$

Step 2.7

Move the limit inside the trig function because cosecant is continuous.

$\frac{1}{- 2} \cdot \frac{- \sec^{2} (lim_{x \to \frac{π}{4}} x)}{\csc^{2} (lim_{x \to \frac{π}{4}} 2 x)}$

Step 2.8

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{- 2} \cdot \frac{- \sec^{2} (lim_{x \to \frac{π}{4}} x)}{\csc^{2} (2 lim_{x \to \frac{π}{4}} x)}$

Step 3

Evaluate the limits by plugging in $\frac{π}{4}$ for all occurrences of $x$ .

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Step 3.1

Evaluate the limit of $x$ by plugging in $\frac{π}{4}$ for $x$ .

$\frac{1}{- 2} \cdot \frac{- \sec^{2} (\frac{π}{4})}{\csc^{2} (2 lim_{x \to \frac{π}{4}} x)}$

Step 3.2

Evaluate the limit of $x$ by plugging in $\frac{π}{4}$ for $x$ .

$\frac{1}{- 2} \cdot \frac{- \sec^{2} (\frac{π}{4})}{\csc^{2} (2 \frac{π}{4})}$

Step 4

Simplify the answer.

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Step 4.1

Combine.

$\frac{1 (- \sec^{2} (\frac{π}{4}))}{- 2 \csc^{2} (2 \frac{π}{4})}$

Step 4.2

Multiply $- 1$ by $1$ .

$\frac{- \sec^{2} (\frac{π}{4})}{- 2 \csc^{2} (2 \frac{π}{4})}$

Step 4.3

Simplify the denominator.

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Step 4.3.1

Cancel the common factor of $2$ .

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Step 4.3.1.1

Factor $2$ out of $4$ .

$\frac{- \sec^{2} (\frac{π}{4})}{- 2 \csc^{2} (2 \frac{π}{2 (2)})}$

Step 4.3.1.2

Cancel the common factor.

$\frac{- \sec^{2} (\frac{π}{4})}{- 2 \csc^{2} (2 \frac{π}{2 \cdot 2})}$

Step 4.3.1.3

Rewrite the expression.

$\frac{- \sec^{2} (\frac{π}{4})}{- 2 \csc^{2} (\frac{π}{2})}$

Step 4.3.2

The exact value of $\csc (\frac{π}{2})$ is $1$ .

$\frac{- \sec^{2} (\frac{π}{4})}{- 2 \cdot 1^{2}}$

Step 4.3.3

One to any power is one.

$\frac{- \sec^{2} (\frac{π}{4})}{- 2 \cdot 1}$

Step 4.4

Simplify the numerator.

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Step 4.4.1

The exact value of $\sec (\frac{π}{4})$ is $\frac{2}{\sqrt{2}}$ .

$\frac{- {(\frac{2}{\sqrt{2}})}^{2}}{- 2 \cdot 1}$

Step 4.4.2

Multiply $\frac{2}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{- {(\frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})}^{2}}{- 2 \cdot 1}$

Step 4.4.3

Combine and simplify the denominator.

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Step 4.4.3.1

Multiply $\frac{2}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{- {(\frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}})}^{2}}{- 2 \cdot 1}$

Step 4.4.3.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{- {(\frac{2 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}})}^{2}}{- 2 \cdot 1}$

Step 4.4.3.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{- {(\frac{2 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}})}^{2}}{- 2 \cdot 1}$

Step 4.4.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{- {(\frac{2 \sqrt{2}}{{\sqrt{2}}^{1 + 1}})}^{2}}{- 2 \cdot 1}$

Step 4.4.3.5

Add $1$ and $1$ .

$\frac{- {(\frac{2 \sqrt{2}}{{\sqrt{2}}^{2}})}^{2}}{- 2 \cdot 1}$

Step 4.4.3.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 4.4.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{- {(\frac{2 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}})}^{2}}{- 2 \cdot 1}$

Step 4.4.3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{- {(\frac{2 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}})}^{2}}{- 2 \cdot 1}$

Step 4.4.3.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{- {(\frac{2 \sqrt{2}}{2^{\frac{2}{2}}})}^{2}}{- 2 \cdot 1}$

Step 4.4.3.6.4

Cancel the common factor of $2$ .

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Step 4.4.3.6.4.1

Cancel the common factor.

$\frac{- {(\frac{2 \sqrt{2}}{2^{\frac{2}{2}}})}^{2}}{- 2 \cdot 1}$

Step 4.4.3.6.4.2

Rewrite the expression.

$\frac{- {(\frac{2 \sqrt{2}}{2^{1}})}^{2}}{- 2 \cdot 1}$

Step 4.4.3.6.5

Evaluate the exponent.

$\frac{- {(\frac{2 \sqrt{2}}{2})}^{2}}{- 2 \cdot 1}$

Step 4.4.4

Cancel the common factor of $2$ .

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Step 4.4.4.1

Cancel the common factor.

$\frac{- {(\frac{2 \sqrt{2}}{2})}^{2}}{- 2 \cdot 1}$

Step 4.4.4.2

Divide $\sqrt{2}$ by $1$ .

$\frac{- {\sqrt{2}}^{2}}{- 2 \cdot 1}$

Step 4.4.5

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 4.4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{- {(2^{\frac{1}{2}})}^{2}}{- 2 \cdot 1}$

Step 4.4.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{- 2^{\frac{1}{2} \cdot 2}}{- 2 \cdot 1}$

Step 4.4.5.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{- 2^{\frac{2}{2}}}{- 2 \cdot 1}$

Step 4.4.5.4

Cancel the common factor of $2$ .

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Step 4.4.5.4.1

Cancel the common factor.

$\frac{- 2^{\frac{2}{2}}}{- 2 \cdot 1}$

Step 4.4.5.4.2

Rewrite the expression.

$\frac{- 2^{1}}{- 2 \cdot 1}$

Step 4.4.5.5

Evaluate the exponent.

$\frac{- 1 \cdot 2}{- 2 \cdot 1}$

Step 4.5

Multiply $- 2$ by $1$ .

$\frac{- 1 \cdot 2}{- 2}$

Step 4.6

Multiply $- 1$ by $2$ .

$\frac{- 2}{- 2}$

Step 4.7

Cancel the common factor of $- 2$ .

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Step 4.7.1

Cancel the common factor.

$\frac{- 2}{- 2}$

Step 4.7.2

Rewrite the expression.

$1$