Calculus Examples

$lim_{x \to 0} \frac{e^{x} - \cos (x) - 2 x}{x^{2} - 2 x}$

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.

$\frac{lim_{x \to 0} e^{x} - \cos (x) - 2 x}{lim_{x \to 0} x^{2} - 2 x}$

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 $0$ .

$\frac{lim_{x \to 0} e^{x} - lim_{x \to 0} \cos (x) - lim_{x \to 0} 2 x}{lim_{x \to 0} x^{2} - 2 x}$

Step 1.2.2

Move the limit into the exponent.

$\frac{e^{lim_{x \to 0} x} - lim_{x \to 0} \cos (x) - lim_{x \to 0} 2 x}{lim_{x \to 0} x^{2} - 2 x}$

Step 1.2.3

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

$\frac{e^{lim_{x \to 0} x} - \cos (lim_{x \to 0} x) - lim_{x \to 0} 2 x}{lim_{x \to 0} x^{2} - 2 x}$

Step 1.2.4

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

$\frac{e^{lim_{x \to 0} x} - \cos (lim_{x \to 0} x) - 2 lim_{x \to 0} x}{lim_{x \to 0} x^{2} - 2 x}$

Step 1.2.5

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 1.2.5.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{e^{0} - \cos (lim_{x \to 0} x) - 2 lim_{x \to 0} x}{lim_{x \to 0} x^{2} - 2 x}$

Step 1.2.5.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{e^{0} - \cos (0) - 2 lim_{x \to 0} x}{lim_{x \to 0} x^{2} - 2 x}$

Step 1.2.5.3

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{e^{0} - \cos (0) - 2 \cdot 0}{lim_{x \to 0} x^{2} - 2 x}$

Step 1.2.6

Simplify the answer.

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Step 1.2.6.1

Simplify each term.

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Step 1.2.6.1.1

Anything raised to $0$ is $1$ .

$\frac{1 - \cos (0) - 2 \cdot 0}{lim_{x \to 0} x^{2} - 2 x}$

Step 1.2.6.1.2

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

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

Step 1.2.6.1.3

Multiply $- 1$ by $1$ .

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

Step 1.2.6.1.4

Multiply $- 2$ by $0$ .

$\frac{1 - 1 + 0}{lim_{x \to 0} x^{2} - 2 x}$

Step 1.2.6.2

Subtract $1$ from $1$ .

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

Step 1.2.6.3

Add $0$ and $0$ .

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

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 1.3.4.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

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

Step 1.3.4.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{0}{0^{2} - 2 \cdot 0}$

Step 1.3.5

Simplify the answer.

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Step 1.3.5.1

Simplify each term.

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Step 1.3.5.1.1

Raising $0$ to any positive power yields $0$ .

$\frac{0}{0 - 2 \cdot 0}$

Step 1.3.5.1.2

Multiply $- 2$ by $0$ .

$\frac{0}{0 + 0}$

Step 1.3.5.2

Add $0$ and $0$ .

$\frac{0}{0}$

Step 1.3.5.3

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

Undefined

$\frac{0}{0}$

Step 1.3.6

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.

$lim_{x \to 0} \frac{e^{x} - \cos (x) - 2 x}{x^{2} - 2 x} = lim_{x \to 0} \frac{\frac{d}{d x} [e^{x} - \cos (x) - 2 x]}{\frac{d}{d x} [x^{2} - 2 x]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [e^{x} - \cos (x) - 2 x]}{\frac{d}{d x} [x^{2} - 2 x]}$

Step 3.2

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

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

Step 3.3

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

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

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

Multiply $- 1$ by $- 1$ .

$lim_{x \to 0} \frac{e^{x} + 1 \sin (x) + \frac{d}{d x} [- 2 x]}{\frac{d}{d x} [x^{2} - 2 x]}$

Step 3.4.4

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

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

Step 3.5

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

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Step 3.5.1

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

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

Step 3.5.2

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

$lim_{x \to 0} \frac{e^{x} + \sin (x) - 2 \cdot 1}{\frac{d}{d x} [x^{2} - 2 x]}$

Step 3.5.3

Multiply $- 2$ by $1$ .

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

Step 3.6

Reorder terms.

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

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

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Step 3.9.1

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

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

Step 3.9.2

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

$lim_{x \to 0} \frac{e^{x} - 2 + \sin (x)}{2 x - 2 \cdot 1}$

Step 3.9.3

Multiply $- 2$ by $1$ .

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

Step 4

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $0$ .

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

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

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

Step 6

Move the limit into the exponent.

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

Step 7

Evaluate the limit of $2$ which is constant as $x$ approaches $0$ .

$\frac{e^{lim_{x \to 0} x} - 1 \cdot 2 + lim_{x \to 0} \sin (x)}{lim_{x \to 0} 2 x - 2}$

Step 8

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

$\frac{e^{lim_{x \to 0} x} - 1 \cdot 2 + \sin (lim_{x \to 0} x)}{lim_{x \to 0} 2 x - 2}$

Step 9

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{e^{lim_{x \to 0} x} - 1 \cdot 2 + \sin (lim_{x \to 0} x)}{lim_{x \to 0} 2 x - lim_{x \to 0} 2}$

Step 10

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

$\frac{e^{lim_{x \to 0} x} - 1 \cdot 2 + \sin (lim_{x \to 0} x)}{2 lim_{x \to 0} x - lim_{x \to 0} 2}$

Step 11

Evaluate the limit of $2$ which is constant as $x$ approaches $0$ .

$\frac{e^{lim_{x \to 0} x} - 1 \cdot 2 + \sin (lim_{x \to 0} x)}{2 lim_{x \to 0} x - 1 \cdot 2}$

Step 12

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 12.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{e^{0} - 1 \cdot 2 + \sin (lim_{x \to 0} x)}{2 lim_{x \to 0} x - 1 \cdot 2}$

Step 12.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{e^{0} - 1 \cdot 2 + \sin (0)}{2 lim_{x \to 0} x - 1 \cdot 2}$

Step 12.3

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{e^{0} - 1 \cdot 2 + \sin (0)}{2 \cdot 0 - 1 \cdot 2}$

Step 13

Simplify the answer.

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Step 13.1

Simplify the numerator.

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Step 13.1.1

Anything raised to $0$ is $1$ .

$\frac{1 - 1 \cdot 2 + \sin (0)}{2 \cdot 0 - 1 \cdot 2}$

Step 13.1.2

Multiply $- 1$ by $2$ .

$\frac{1 - 2 + \sin (0)}{2 \cdot 0 - 1 \cdot 2}$

Step 13.1.3

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

$\frac{1 - 2 + 0}{2 \cdot 0 - 1 \cdot 2}$

Step 13.1.4

Add $1 - 2$ and $0$ .

$\frac{1 - 2}{2 \cdot 0 - 1 \cdot 2}$

Step 13.1.5

Subtract $2$ from $1$ .

$\frac{- 1}{2 \cdot 0 - 1 \cdot 2}$

Step 13.2

Simplify the denominator.

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Step 13.2.1

Multiply $2$ by $0$ .

$\frac{- 1}{0 - 1 \cdot 2}$

Step 13.2.2

Multiply $- 1$ by $2$ .

$\frac{- 1}{0 - 2}$

Step 13.2.3

Subtract $2$ from $0$ .

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

Step 13.3

Dividing two negative values results in a positive value.

$\frac{1}{2}$