Calculus Examples

$lim_{x \to 0} \frac{\sin (2 x)}{\cos (5 x)}$

Step 1

Apply trigonometric identities.

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Step 1.1

Rewrite $\frac{\sin (2 x)}{\cos (5 x)}$ as a product.

$lim_{x \to 0} \sin (2 x) \frac{1}{\cos (5 x)}$

Step 1.2

Write $\sin (2 x)$ as a fraction with denominator $1$ .

$lim_{x \to 0} \frac{\sin (2 x)}{1} \cdot \frac{1}{\cos (5 x)}$

Step 1.3

Simplify.

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Step 1.3.1

Divide $\sin (2 x)$ by $1$ .

$lim_{x \to 0} \sin (2 x) \frac{1}{\cos (5 x)}$

Step 1.3.2

Convert from $\frac{1}{\cos (5 x)}$ to $\sec (5 x)$ .

$lim_{x \to 0} \sin (2 x) \sec (5 x)$

Step 2

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

$lim_{x \to 0} \sin (2 x) \cdot lim_{x \to 0} \sec (5 x)$

Step 3

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

$\sin (lim_{x \to 0} 2 x) \cdot lim_{x \to 0} \sec (5 x)$

Step 4

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

$\sin (2 lim_{x \to 0} x) \cdot lim_{x \to 0} \sec (5 x)$

Step 5

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

$\sin (2 lim_{x \to 0} x) \cdot \sec (lim_{x \to 0} 5 x)$

Step 6

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\sin (2 lim_{x \to 0} x) \cdot \sec (5 lim_{x \to 0} x)$

Step 7

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

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Step 7.1

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

$\sin (2 \cdot 0) \cdot \sec (5 lim_{x \to 0} x)$

Step 7.2

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

$\sin (2 \cdot 0) \cdot \sec (5 \cdot 0)$

Step 8

Simplify the answer.

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Step 8.1

Multiply $2$ by $0$ .

$\sin (0) \cdot \sec (5 \cdot 0)$

Step 8.2

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

$0 \cdot \sec (5 \cdot 0)$

Step 8.3

Multiply $5$ by $0$ .

$0 \cdot \sec (0)$

Step 8.4

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

$0 \cdot 1$

Step 8.5

Multiply $0$ by $1$ .

$0$