Calculus Examples

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

Step 1

Combine terms.

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Step 1.1

To write $x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$lim_{x \to \frac{π}{2}} \frac{\tan (2 π)}{x \cdot \frac{2}{2} - \frac{π}{2}}$

Step 1.2

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

$lim_{x \to \frac{π}{2}} \frac{\tan (2 π)}{\frac{x \cdot 2}{2} - \frac{π}{2}}$

Step 1.3

Combine the numerators over the common denominator.

$lim_{x \to \frac{π}{2}} \frac{\tan (2 π)}{\frac{x \cdot 2 - π}{2}}$

Step 2

Simplify the limit argument.

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Step 2.1

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to \frac{π}{2}} \tan (2 π) \frac{2}{x \cdot 2 - π}$

Step 2.2

Combine $\tan (2 π)$ and $\frac{2}{x \cdot 2 - π}$ .

$lim_{x \to \frac{π}{2}} \frac{\tan (2 π) \cdot 2}{x \cdot 2 - π}$

Step 3

Apply L'Hospital's rule.

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Step 3.1

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

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Step 3.1.1

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

$\frac{lim_{x \to \frac{π}{2}} \tan (2 π) \cdot 2}{lim_{x \to \frac{π}{2}} x \cdot 2 - π}$

Step 3.1.2

Evaluate the limit of the numerator.

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Step 3.1.2.1

Evaluate the limit of $\tan (2 π) \cdot 2$ which is constant as $x$ approaches $\frac{π}{2}$ .

$\frac{\tan (2 π) \cdot 2}{lim_{x \to \frac{π}{2}} x \cdot 2 - π}$

Step 3.1.2.2

Simplify the answer.

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Step 3.1.2.2.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\frac{\tan (0) \cdot 2}{lim_{x \to \frac{π}{2}} x \cdot 2 - π}$

Step 3.1.2.2.2

The exact value of $\tan (0)$ is $0$ .

$\frac{0 \cdot 2}{lim_{x \to \frac{π}{2}} x \cdot 2 - π}$

Step 3.1.2.2.3

Multiply $0$ by $2$ .

$\frac{0}{lim_{x \to \frac{π}{2}} x \cdot 2 - π}$

Step 3.1.3

Evaluate the limit of the denominator.

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Step 3.1.3.1

Evaluate the limit.

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Step 3.1.3.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{π}{2}$ .

$\frac{0}{lim_{x \to \frac{π}{2}} x \cdot 2 - lim_{x \to \frac{π}{2}} π}$

Step 3.1.3.1.2

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

$\frac{0}{2 lim_{x \to \frac{π}{2}} x - lim_{x \to \frac{π}{2}} π}$

Step 3.1.3.1.3

Evaluate the limit of $π$ which is constant as $x$ approaches $\frac{π}{2}$ .

$\frac{0}{2 lim_{x \to \frac{π}{2}} x - π}$

Step 3.1.3.2

Evaluate the limit of $x$ by plugging in $\frac{π}{2}$ for $x$ .

$\frac{0}{2 \frac{π}{2} - π}$

Step 3.1.3.3

Simplify the answer.

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Step 3.1.3.3.1

Cancel the common factor of $2$ .

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Step 3.1.3.3.1.1

Cancel the common factor.

$\frac{0}{2 \frac{π}{2} - π}$

Step 3.1.3.3.1.2

Rewrite the expression.

$\frac{0}{π - π}$

Step 3.1.3.3.2

Subtract $π$ from $π$ .

$\frac{0}{0}$

Step 3.1.3.3.3

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

Undefined

$\frac{0}{0}$

Step 3.1.3.4

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

Undefined

$\frac{0}{0}$

Step 3.1.4

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

Undefined

$\frac{0}{0}$

Step 3.2

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.

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

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

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

Step 3.3.2

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

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

Step 3.3.3

The exact value of $\tan (0)$ is $0$ .

$lim_{x \to \frac{π}{2}} \frac{\frac{d}{d x} [0 \cdot 2]}{\frac{d}{d x} [x \cdot 2 - π]}$

Step 3.3.4

Multiply $0$ by $2$ .

$lim_{x \to \frac{π}{2}} \frac{\frac{d}{d x} [0]}{\frac{d}{d x} [x \cdot 2 - π]}$

Step 3.3.5

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

$lim_{x \to \frac{π}{2}} \frac{0}{\frac{d}{d x} [x \cdot 2 - π]}$

Step 3.3.6

By the Sum Rule, the derivative of $x \cdot 2 - π$ with respect to $x$ is $\frac{d}{d x} [x \cdot 2] + \frac{d}{d x} [- π]$ .

$lim_{x \to \frac{π}{2}} \frac{0}{\frac{d}{d x} [x \cdot 2] + \frac{d}{d x} [- π]}$

Step 3.3.7

Evaluate $\frac{d}{d x} [x \cdot 2]$ .

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Step 3.3.7.1

Move $2$ to the left of $x$ .

$lim_{x \to \frac{π}{2}} \frac{0}{\frac{d}{d x} [2 \cdot x] + \frac{d}{d x} [- π]}$

Step 3.3.7.2

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

$lim_{x \to \frac{π}{2}} \frac{0}{2 \frac{d}{d x} [x] + \frac{d}{d x} [- π]}$

Step 3.3.7.3

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

$lim_{x \to \frac{π}{2}} \frac{0}{2 \cdot 1 + \frac{d}{d x} [- π]}$

Step 3.3.7.4

Multiply $2$ by $1$ .

$lim_{x \to \frac{π}{2}} \frac{0}{2 + \frac{d}{d x} [- π]}$

Step 3.3.8

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

$lim_{x \to \frac{π}{2}} \frac{0}{2 + 0}$

Step 3.3.9

Add $2$ and $0$ .

$lim_{x \to \frac{π}{2}} \frac{0}{2}$

Step 3.4

Cancel the common factor of $0$ and $2$ .

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Step 3.4.1

Factor $2$ out of $0$ .

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

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Factor $2$ out of $2$ .

$lim_{x \to \frac{π}{2}} \frac{2 \cdot 0}{2 \cdot 1}$

Step 3.4.2.2

Cancel the common factor.

$lim_{x \to \frac{π}{2}} \frac{2 \cdot 0}{2 \cdot 1}$

Step 3.4.2.3

Rewrite the expression.

$lim_{x \to \frac{π}{2}} \frac{0}{1}$

Step 3.4.2.4

Divide $0$ by $1$ .

$lim_{x \to \frac{π}{2}} 0$

Step 4

Evaluate the limit of $0$ which is constant as $x$ approaches $\frac{π}{2}$ .

$0$