Calculus Examples

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

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.

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

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 $\frac{π}{4}$ .

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

Step 1.1.2.1.2

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

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

Step 1.1.2.1.3

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

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

Step 1.1.2.2

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

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

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 $1$ .

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

Step 1.1.2.3.1.2

Multiply $- 1$ by $1$ .

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

Step 1.1.2.3.2

Subtract $1$ from $1$ .

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

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

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

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

Step 1.1.3.1.2

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

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

Step 1.1.3.1.3

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

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

Step 1.1.3.2

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

$\frac{0}{4 \frac{π}{4} - π}$

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Cancel the common factor of $4$ .

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Step 1.1.3.3.1.1

Cancel the common factor.

$\frac{0}{4 \frac{π}{4} - π}$

Step 1.1.3.3.1.2

Rewrite the expression.

$\frac{0}{π - π}$

Step 1.1.3.3.2

Subtract $π$ from $π$ .

$\frac{0}{0}$

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.

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

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

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

Step 1.3.5

Simplify.

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Step 1.3.5.1

Add $\sec^{2} (x)$ and $0$ .

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

Step 1.3.5.2

Rewrite $\sec (x)$ in terms of sines and cosines.

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

Step 1.3.5.3

Apply the product rule to $\frac{1}{\cos (x)}$ .

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

Step 1.3.5.4

One to any power is one.

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

Step 1.3.6

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

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

Step 1.3.7

Evaluate $\frac{d}{d x} [4 x]$ .

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Step 1.3.7.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

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

Step 1.3.7.2

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

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

Step 1.3.7.3

Multiply $4$ by $1$ .

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

Step 1.3.8

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

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

Step 1.3.9

Add $4$ and $0$ .

$lim_{x \to \frac{π}{4}} \frac{\frac{1}{\cos^{2} (x)}}{4}$

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.5

Multiply $\frac{1}{\cos^{2} (x)}$ by $\frac{1}{4}$ .

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

Step 2

Evaluate the limit.

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Step 2.1

Move the term $\frac{1}{4}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{4} lim_{x \to \frac{π}{4}} \frac{1}{\cos^{2} (x)}$

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 3

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

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

Step 4

Simplify the answer.

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Step 4.1

Combine.

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

Step 4.2

Multiply $1$ by $1$ .

$\frac{1}{4 \cos^{2} (\frac{π}{4})}$

Step 4.3

Simplify the denominator.

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Step 4.3.1

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

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

Step 4.3.2

Apply the product rule to $\frac{\sqrt{2}}{2}$ .

$\frac{1}{4 \frac{{\sqrt{2}}^{2}}{2^{2}}}$

Step 4.3.3

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

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Step 4.3.3.1

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

$\frac{1}{4 \frac{{(2^{\frac{1}{2}})}^{2}}{2^{2}}}$

Step 4.3.3.2

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

$\frac{1}{4 \frac{2^{\frac{1}{2} \cdot 2}}{2^{2}}}$

Step 4.3.3.3

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

$\frac{1}{4 \frac{2^{\frac{2}{2}}}{2^{2}}}$

Step 4.3.3.4

Cancel the common factor of $2$ .

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Step 4.3.3.4.1

Cancel the common factor.

$\frac{1}{4 \frac{2^{\frac{2}{2}}}{2^{2}}}$

Step 4.3.3.4.2

Rewrite the expression.

$\frac{1}{4 \frac{2^{1}}{2^{2}}}$

Step 4.3.3.5

Evaluate the exponent.

$\frac{1}{4 \frac{2}{2^{2}}}$

Step 4.3.4

Raise $2$ to the power of $2$ .

$\frac{1}{4 (\frac{2}{4})}$

Step 4.3.5

Cancel the common factor of $2$ and $4$ .

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Step 4.3.5.1

Factor $2$ out of $2$ .

$\frac{1}{4 \frac{2 (1)}{4}}$

Step 4.3.5.2

Cancel the common factors.

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Step 4.3.5.2.1

Factor $2$ out of $4$ .

$\frac{1}{4 \frac{2 \cdot 1}{2 \cdot 2}}$

Step 4.3.5.2.2

Cancel the common factor.

$\frac{1}{4 \frac{2 \cdot 1}{2 \cdot 2}}$

Step 4.3.5.2.3

Rewrite the expression.

$\frac{1}{4 (\frac{1}{2})}$

Step 4.4

Combine $4$ and $\frac{1}{2}$ .

$\frac{1}{\frac{4}{2}}$

Step 4.5

Divide $4$ by $2$ .

$\frac{1}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$