Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

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

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Multiply $3$ by $0$ .

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

Step 3.1.2

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

$\frac{2 \cdot 1 - 1 \cdot 2}{3 x \sec (3 x)}$

Step 3.1.3

Multiply $2$ by $1$ .

$\frac{2 - 1 \cdot 2}{3 x \sec (3 x)}$

Step 3.1.4

Multiply $- 1$ by $2$ .

$\frac{2 - 2}{3 x \sec (3 x)}$

Step 3.1.5

Subtract $2$ from $2$ .

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

Step 3.2

Cancel the common factor of $0$ and $3$ .

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Step 3.2.1

Factor $3$ out of $0$ .

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

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

Factor $3$ out of $3 x \sec (3 x)$ .

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

Step 3.2.2.2

Cancel the common factor.

$\frac{3 \cdot 0}{3 (x \sec (3 x))}$

Step 3.2.2.3

Rewrite the expression.

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

Step 3.3

Separate fractions.

$\frac{0}{x} \cdot \frac{1}{\sec (3 x)}$

Step 3.4

Rewrite $\sec (3 x)$ in terms of sines and cosines.

$\frac{0}{x} \cdot \frac{1}{\frac{1}{\cos (3 x)}}$

Step 3.5

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (3 x)}$ .

$\frac{0}{x} (1 \cos (3 x))$

Step 3.6

Multiply $\cos (3 x)$ by $1$ .

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

Step 3.7

Divide $0$ by $x$ .

$0 \cos (3 x)$

Step 3.8

Multiply $0$ by $\cos (3 x)$ .

$0$