Calculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

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

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Step 1.2.4.1

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

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

Step 1.2.4.2

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

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

Step 1.2.5

Simplify the answer.

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Step 1.2.5.1

Simplify each term.

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Step 1.2.5.1.1

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

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

Step 1.2.5.1.2

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

$\frac{1 - 1 \cdot 1}{lim_{x \to \frac{π}{4}} x - \frac{π}{4}}$

Step 1.2.5.1.3

Multiply $- 1$ by $1$ .

$\frac{1 - 1}{lim_{x \to \frac{π}{4}} x - \frac{π}{4}}$

Step 1.2.5.2

Subtract $1$ from $1$ .

$\frac{0}{lim_{x \to \frac{π}{4}} x - \frac{π}{4}}$

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

Evaluate the limit.

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Step 1.3.1.1

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

$\frac{0}{lim_{x \to \frac{π}{4}} x - lim_{x \to \frac{π}{4}} \frac{π}{4}}$

Step 1.3.1.2

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

$\frac{0}{lim_{x \to \frac{π}{4}} x - \frac{π}{4}}$

Step 1.3.2

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

$\frac{0}{\frac{π}{4} - \frac{π}{4}}$

Step 1.3.3

Simplify the answer.

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Step 1.3.3.1

Combine the numerators over the common denominator.

$\frac{0}{\frac{π - π}{4}}$

Step 1.3.3.2

Subtract $π$ from $π$ .

$\frac{0}{\frac{0}{4}}$

Step 1.3.3.3

Divide $0$ by $4$ .

$\frac{0}{0}$

Step 1.3.3.4

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

Undefined

$\frac{0}{0}$

Step 1.3.4

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

Undefined

$\frac{0}{0}$

Step 1.4

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

Undefined

$\frac{0}{0}$

Step 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.

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

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

Multiply $- 1$ by $- 1$ .

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

Step 3.4.4

Multiply $\csc^{2} (x)$ by $1$ .

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

Step 3.5

By the Sum Rule, the derivative of $x - \frac{π}{4}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{π}{4}]$ .

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

Step 3.6

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

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

Step 3.7

Since $- \frac{π}{4}$ is constant with respect to $x$ , the derivative of $- \frac{π}{4}$ with respect to $x$ is $0$ .

$lim_{x \to \frac{π}{4}} \frac{\sec^{2} (x) + \csc^{2} (x)}{1 + 0}$

Step 3.8

Add $1$ and $0$ .

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

Step 4

Divide $\sec^{2} (x) + \csc^{2} (x)$ by $1$ .

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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Step 10.1

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

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

Step 10.2

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

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

Step 11

Simplify the answer.

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Step 11.1

Simplify each term.

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Step 11.1.1

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

${(\frac{2}{\sqrt{2}})}^{2} + \csc^{2} (\frac{π}{4})$

Step 11.1.2

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

${(\frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})}^{2} + \csc^{2} (\frac{π}{4})$

Step 11.1.3

Combine and simplify the denominator.

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Step 11.1.3.1

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

${(\frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}})}^{2} + \csc^{2} (\frac{π}{4})$

Step 11.1.3.2

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

${(\frac{2 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}})}^{2} + \csc^{2} (\frac{π}{4})$

Step 11.1.3.3

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

${(\frac{2 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}})}^{2} + \csc^{2} (\frac{π}{4})$

Step 11.1.3.4

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

${(\frac{2 \sqrt{2}}{{\sqrt{2}}^{1 + 1}})}^{2} + \csc^{2} (\frac{π}{4})$

Step 11.1.3.5

Add $1$ and $1$ .

${(\frac{2 \sqrt{2}}{{\sqrt{2}}^{2}})}^{2} + \csc^{2} (\frac{π}{4})$

Step 11.1.3.6

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

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Step 11.1.3.6.1

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

${(\frac{2 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}})}^{2} + \csc^{2} (\frac{π}{4})$

Step 11.1.3.6.2

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

${(\frac{2 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}})}^{2} + \csc^{2} (\frac{π}{4})$

Step 11.1.3.6.3

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

${(\frac{2 \sqrt{2}}{2^{\frac{2}{2}}})}^{2} + \csc^{2} (\frac{π}{4})$

Step 11.1.3.6.4

Cancel the common factor of $2$ .

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Step 11.1.3.6.4.1

Cancel the common factor.

${(\frac{2 \sqrt{2}}{2^{\frac{2}{2}}})}^{2} + \csc^{2} (\frac{π}{4})$

Step 11.1.3.6.4.2

Rewrite the expression.

${(\frac{2 \sqrt{2}}{2^{1}})}^{2} + \csc^{2} (\frac{π}{4})$

Step 11.1.3.6.5

Evaluate the exponent.

${(\frac{2 \sqrt{2}}{2})}^{2} + \csc^{2} (\frac{π}{4})$

Step 11.1.4

Cancel the common factor of $2$ .

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Step 11.1.4.1

Cancel the common factor.

${(\frac{2 \sqrt{2}}{2})}^{2} + \csc^{2} (\frac{π}{4})$

Step 11.1.4.2

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

${\sqrt{2}}^{2} + \csc^{2} (\frac{π}{4})$

Step 11.1.5

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

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Step 11.1.5.1

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

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

Step 11.1.5.2

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

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

Step 11.1.5.3

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

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

Step 11.1.5.4

Cancel the common factor of $2$ .

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Step 11.1.5.4.1

Cancel the common factor.

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

Step 11.1.5.4.2

Rewrite the expression.

$2^{1} + \csc^{2} (\frac{π}{4})$

Step 11.1.5.5

Evaluate the exponent.

$2 + \csc^{2} (\frac{π}{4})$

Step 11.1.6

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

$2 + {\sqrt{2}}^{2}$

Step 11.1.7

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

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Step 11.1.7.1

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

$2 + {(2^{\frac{1}{2}})}^{2}$

Step 11.1.7.2

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

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

Step 11.1.7.3

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

$2 + 2^{\frac{2}{2}}$

Step 11.1.7.4

Cancel the common factor of $2$ .

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Step 11.1.7.4.1

Cancel the common factor.

$2 + 2^{\frac{2}{2}}$

Step 11.1.7.4.2

Rewrite the expression.

$2 + 2^{1}$

Step 11.1.7.5

Evaluate the exponent.

$2 + 2$

Step 11.2

Add $2$ and $2$ .

$4$