Calculus Examples

$lim_{x \to 0} \frac{9 \tan (x)}{- 3 x}$

Step 1

Evaluate the limit.

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Step 1.1

Cancel the common factor of $9$ and $- 3$ .

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Step 1.1.1

Factor $3$ out of $9 \tan (x)$ .

$lim_{x \to 0} \frac{3 (3 \tan (x))}{- 3 x}$

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

Factor $3$ out of $- 3 x$ .

$lim_{x \to 0} \frac{3 (3 \tan (x))}{3 (- x)}$

Step 1.1.2.2

Cancel the common factor.

$lim_{x \to 0} \frac{3 (3 \tan (x))}{3 (- x)}$

Step 1.1.2.3

Rewrite the expression.

$lim_{x \to 0} \frac{3 \tan (x)}{- x}$

Step 1.2

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$3 lim_{x \to 0} \frac{\tan (x)}{- x}$

Step 2

Apply L'Hospital's rule.

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Step 2.1

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

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Step 2.1.1

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

$3 \frac{lim_{x \to 0} \tan (x)}{lim_{x \to 0} - x}$

Step 2.1.2

Evaluate the limit of the numerator.

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Step 2.1.2.1

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

$3 \frac{\tan (lim_{x \to 0} x)}{lim_{x \to 0} - x}$

Step 2.1.2.2

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

$3 \frac{\tan (0)}{lim_{x \to 0} - x}$

Step 2.1.2.3

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

$3 \frac{0}{lim_{x \to 0} - x}$

Step 2.1.3

Evaluate the limit of the denominator.

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Step 2.1.3.1

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

$3 \frac{0}{- lim_{x \to 0} x}$

Step 2.1.3.2

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

$3 (\frac{0}{0})$

Step 2.1.3.3

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

Undefined

$3 (\frac{0}{0})$

Step 2.1.4

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

Undefined

$3 (\frac{0}{0})$

Step 2.2

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.

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

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$3 lim_{x \to 0} \frac{\frac{d}{d x} [\tan (x)]}{\frac{d}{d x} [- x]}$

Step 2.3.2

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

$3 lim_{x \to 0} \frac{\sec^{2} (x)}{\frac{d}{d x} [- x]}$

Step 2.3.3

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

$3 lim_{x \to 0} \frac{\sec^{2} (x)}{- \frac{d}{d x} [x]}$

Step 2.3.4

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

$3 lim_{x \to 0} \frac{\sec^{2} (x)}{- 1 \cdot 1}$

Step 2.3.5

Multiply $- 1$ by $1$ .

$3 lim_{x \to 0} \frac{\sec^{2} (x)}{- 1}$

Step 2.4

Move the negative one from the denominator of $\frac{\sec^{2} (x)}{- 1}$ .

$3 lim_{x \to 0} - 1 \cdot \sec^{2} (x)$

Step 3

Evaluate the limit.

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Step 3.1

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

$3 \cdot - 1 lim_{x \to 0} \sec^{2} (x)$

Step 3.2

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

$3 \cdot - 1 {(lim_{x \to 0} \sec (x))}^{2}$

Step 3.3

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

$3 \cdot - 1 \sec^{2} (lim_{x \to 0} x)$

Step 4

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

$3 \cdot - 1 \sec^{2} (0)$

Step 5

Simplify the answer.

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Step 5.1

Multiply $3$ by $- 1$ .

$- 3 \sec^{2} (0)$

Step 5.2

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

$- 3 \cdot 1^{2}$

Step 5.3

One to any power is one.

$- 3 \cdot 1$

Step 5.4

Multiply $- 3$ by $1$ .

$- 3$