Calculus Examples

$lim_{x \to 0} \frac{\sin (x) + x}{x}$

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.

$\frac{lim_{x \to 0} \sin (x) + x}{lim_{x \to 0} x}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

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

$\frac{lim_{x \to 0} \sin (x) + lim_{x \to 0} x}{lim_{x \to 0} x}$

Step 1.1.2.2

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

$\frac{\sin (lim_{x \to 0} x) + lim_{x \to 0} x}{lim_{x \to 0} x}$

Step 1.1.2.3

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

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Step 1.1.2.3.1

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

$\frac{\sin (0) + lim_{x \to 0} x}{lim_{x \to 0} x}$

Step 1.1.2.3.2

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

$\frac{\sin (0) + 0}{lim_{x \to 0} x}$

Step 1.1.2.4

Simplify the answer.

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Step 1.1.2.4.1

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

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

Step 1.1.2.4.2

Add $0$ and $0$ .

$\frac{0}{lim_{x \to 0} x}$

Step 1.1.3

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

$\frac{0}{0}$

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.

$lim_{x \to 0} \frac{\sin (x) + x}{x} = lim_{x \to 0} \frac{\frac{d}{d x} [\sin (x) + x]}{\frac{d}{d x} [x]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

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

$lim_{x \to 0} \frac{\cos (x) + 1}{\frac{d}{d x} [x]}$

Step 1.3.5

Reorder terms.

$lim_{x \to 0} \frac{1 + \cos (x)}{\frac{d}{d x} [x]}$

Step 1.3.6

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

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

Step 1.4

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

$lim_{x \to 0} 1 + \cos (x)$

Step 2

Evaluate the limit.

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Step 2.1

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

$lim_{x \to 0} 1 + lim_{x \to 0} \cos (x)$

Step 2.2

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$1 + lim_{x \to 0} \cos (x)$

Step 2.3

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

$1 + \cos (lim_{x \to 0} x)$

Step 3

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

$1 + \cos (0)$

Step 4

Simplify the answer.

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Step 4.1

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

$1 + 1$

Step 4.2

Add $1$ and $1$ .

$2$