Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 5

Simplify the answer.

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Step 5.1

Separate fractions.

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

Step 5.2

Convert from $\frac{1}{\cos (0)}$ to $\sec (0)$ .

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

Step 5.3

Divide $0$ by $1$ .

$3 (0 \sec (0))$

Step 5.4

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

$3 (0 \cdot 1)$

Step 5.5

Multiply $3 (0 \cdot 1)$ .

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Step 5.5.1

Multiply $0$ by $1$ .

$3 \cdot 0$

Step 5.5.2

Multiply $3$ by $0$ .

$0$