Calculus Examples

$lim_{x \to 2 π} \frac{x \sin (x) + x^{2} - 4 π^{2}}{x - 2 π}$

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.

$\frac{lim_{x \to 2 π} x \sin (x) + x^{2} - 4 π^{2}}{lim_{x \to 2 π} x - 2 π}$

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 $2 π$ .

$\frac{lim_{x \to 2 π} x \sin (x) + lim_{x \to 2 π} x^{2} - lim_{x \to 2 π} 4 π^{2}}{lim_{x \to 2 π} x - 2 π}$

Step 1.2.2

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $2 π$ .

$\frac{lim_{x \to 2 π} x \cdot lim_{x \to 2 π} \sin (x) + lim_{x \to 2 π} x^{2} - lim_{x \to 2 π} 4 π^{2}}{lim_{x \to 2 π} x - 2 π}$

Step 1.2.3

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

$\frac{lim_{x \to 2 π} x \cdot \sin (lim_{x \to 2 π} x) + lim_{x \to 2 π} x^{2} - lim_{x \to 2 π} 4 π^{2}}{lim_{x \to 2 π} x - 2 π}$

Step 1.2.4

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

$\frac{lim_{x \to 2 π} x \cdot \sin (lim_{x \to 2 π} x) + {(lim_{x \to 2 π} x)}^{2} - lim_{x \to 2 π} 4 π^{2}}{lim_{x \to 2 π} x - 2 π}$

Step 1.2.5

Evaluate the limit of $4 π^{2}$ which is constant as $x$ approaches $2 π$ .

$\frac{lim_{x \to 2 π} x \cdot \sin (lim_{x \to 2 π} x) + {(lim_{x \to 2 π} x)}^{2} - 4 π^{2}}{lim_{x \to 2 π} x - 2 π}$

Step 1.2.6

Evaluate the limits by plugging in $2 π$ for all occurrences of $x$ .

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Step 1.2.6.1

Evaluate the limit of $x$ by plugging in $2 π$ for $x$ .

$\frac{2 π \cdot \sin (lim_{x \to 2 π} x) + {(lim_{x \to 2 π} x)}^{2} - 4 π^{2}}{lim_{x \to 2 π} x - 2 π}$

Step 1.2.6.2

Evaluate the limit of $x$ by plugging in $2 π$ for $x$ .

$\frac{2 π \cdot \sin (2 π) + {(lim_{x \to 2 π} x)}^{2} - 4 π^{2}}{lim_{x \to 2 π} x - 2 π}$

Step 1.2.6.3

Evaluate the limit of $x$ by plugging in $2 π$ for $x$ .

$\frac{2 π \cdot \sin (2 π) + {(2 π)}^{2} - 4 π^{2}}{lim_{x \to 2 π} x - 2 π}$

Step 1.2.7

Simplify the answer.

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Step 1.2.7.1

Simplify each term.

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Step 1.2.7.1.1

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

$\frac{2 π \cdot \sin (0) + {(2 π)}^{2} - 4 π^{2}}{lim_{x \to 2 π} x - 2 π}$

Step 1.2.7.1.2

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

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

Step 1.2.7.1.3

Multiply $2 π \cdot 0$ .

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Step 1.2.7.1.3.1

Multiply $0$ by $2$ .

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

Step 1.2.7.1.3.2

Multiply $0$ by $π$ .

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

Step 1.2.7.1.4

Apply the product rule to $2 π$ .

$\frac{0 + 2^{2} π^{2} - 4 π^{2}}{lim_{x \to 2 π} x - 2 π}$

Step 1.2.7.1.5

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

$\frac{0 + 4 π^{2} - 4 π^{2}}{lim_{x \to 2 π} x - 2 π}$

Step 1.2.7.2

Add $0$ and $4 π^{2}$ .

$\frac{4 π^{2} - 4 π^{2}}{lim_{x \to 2 π} x - 2 π}$

Step 1.2.7.3

Subtract $4 π^{2}$ from $4 π^{2}$ .

$\frac{0}{lim_{x \to 2 π} x - 2 π}$

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 $2 π$ .

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

Step 1.3.1.2

Evaluate the limit of $2 π$ which is constant as $x$ approaches $2 π$ .

$\frac{0}{lim_{x \to 2 π} x - 2 π}$

Step 1.3.2

Evaluate the limit of $x$ by plugging in $2 π$ for $x$ .

$\frac{0}{2 π - 2 π}$

Step 1.3.3

Subtract $2 π$ from $2 π$ .

$\frac{0}{0}$

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.

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

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

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

Step 3.2

By the Sum Rule, the derivative of $x \sin (x) + x^{2} - 4 π^{2}$ with respect to $x$ is $\frac{d}{d x} [x \sin (x)] + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 π^{2}]$ .

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

Step 3.3

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

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Step 3.3.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = \sin (x)$ .

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

Step 3.3.2

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

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

Step 3.3.3

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

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

Step 3.3.4

Multiply $\sin (x)$ by $1$ .

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

Step 3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 3.5

Since $- 4 π^{2}$ is constant with respect to $x$ , the derivative of $- 4 π^{2}$ with respect to $x$ is $0$ .

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

Step 3.6

Simplify.

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Step 3.6.1

Add $x \cos (x) + \sin (x) + 2 x$ and $0$ .

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

Step 3.6.2

Reorder terms.

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

Since $- 2 π$ is constant with respect to $x$ , the derivative of $- 2 π$ with respect to $x$ is $0$ .

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

Step 3.10

Add $1$ and $0$ .

$lim_{x \to 2 π} \frac{x \cos (x) + 2 x + \sin (x)}{1}$

Step 4

Divide $x \cos (x) + 2 x + \sin (x)$ by $1$ .

$lim_{x \to 2 π} x \cos (x) + 2 x + \sin (x)$

Step 5

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

$lim_{x \to 2 π} x \cos (x) + lim_{x \to 2 π} 2 x + lim_{x \to 2 π} \sin (x)$

Step 6

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $2 π$ .

$lim_{x \to 2 π} x \cdot lim_{x \to 2 π} \cos (x) + lim_{x \to 2 π} 2 x + lim_{x \to 2 π} \sin (x)$

Step 7

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

$lim_{x \to 2 π} x \cdot \cos (lim_{x \to 2 π} x) + lim_{x \to 2 π} 2 x + lim_{x \to 2 π} \sin (x)$

Step 8

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

$lim_{x \to 2 π} x \cdot \cos (lim_{x \to 2 π} x) + 2 lim_{x \to 2 π} x + lim_{x \to 2 π} \sin (x)$

Step 9

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

$lim_{x \to 2 π} x \cdot \cos (lim_{x \to 2 π} x) + 2 lim_{x \to 2 π} x + \sin (lim_{x \to 2 π} x)$

Step 10

Evaluate the limits by plugging in $2 π$ for all occurrences of $x$ .

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Step 10.1

Evaluate the limit of $x$ by plugging in $2 π$ for $x$ .

$2 π \cdot \cos (lim_{x \to 2 π} x) + 2 lim_{x \to 2 π} x + \sin (lim_{x \to 2 π} x)$

Step 10.2

Evaluate the limit of $x$ by plugging in $2 π$ for $x$ .

$2 π \cdot \cos (2 π) + 2 lim_{x \to 2 π} x + \sin (lim_{x \to 2 π} x)$

Step 10.3

Evaluate the limit of $x$ by plugging in $2 π$ for $x$ .

$2 π \cdot \cos (2 π) + 2 (2 π) + \sin (lim_{x \to 2 π} x)$

Step 10.4

Evaluate the limit of $x$ by plugging in $2 π$ for $x$ .

$2 π \cdot \cos (2 π) + 2 (2 π) + \sin (2 π)$

Step 11

Simplify the answer.

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Step 11.1

Simplify each term.

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Step 11.1.1

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

$2 π \cdot \cos (0) + 2 (2 π) + \sin (2 π)$

Step 11.1.2

The exact value of $\cos (0)$ is $1$ .

$2 π \cdot 1 + 2 (2 π) + \sin (2 π)$

Step 11.1.3

Multiply $2$ by $1$ .

$2 π + 2 (2 π) + \sin (2 π)$

Step 11.1.4

Multiply $2$ by $2$ .

$2 π + 4 π + \sin (2 π)$

Step 11.1.5

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

$2 π + 4 π + \sin (0)$

Step 11.1.6

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

$2 π + 4 π + 0$

Step 11.2

Add $2 π$ and $4 π$ .

$6 π + 0$

Step 11.3

Add $6 π$ and $0$ .

$6 π$