Calculus Examples

$\frac{lim_{x \to 0} \sin^{2} (5 x^{2}) + \sin^{2} (3 x)}{x^{2}}$

Step 1

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

$\frac{lim_{x \to 0} \sin^{2} (5 x^{2}) + lim_{x \to 0} \sin^{2} (3 x)}{x^{2}}$

Step 2

Move the exponent $2$ from $\sin^{2} (5 x^{2})$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 0} \sin (5 x^{2}))}^{2} + lim_{x \to 0} \sin^{2} (3 x)}{x^{2}}$

Step 3

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

$\frac{\sin^{2} (lim_{x \to 0} 5 x^{2}) + lim_{x \to 0} \sin^{2} (3 x)}{x^{2}}$

Step 4

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

$\frac{\sin^{2} (5 lim_{x \to 0} x^{2}) + lim_{x \to 0} \sin^{2} (3 x)}{x^{2}}$

Step 5

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

$\frac{\sin^{2} (5 {(lim_{x \to 0} x)}^{2}) + lim_{x \to 0} \sin^{2} (3 x)}{x^{2}}$

Step 6

Move the exponent $2$ from $\sin^{2} (3 x)$ outside the limit using the Limits Power Rule.

$\frac{\sin^{2} (5 {(lim_{x \to 0} x)}^{2}) + {(lim_{x \to 0} \sin (3 x))}^{2}}{x^{2}}$

Step 7

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

$\frac{\sin^{2} (5 {(lim_{x \to 0} x)}^{2}) + \sin^{2} (lim_{x \to 0} 3 x)}{x^{2}}$

Step 8

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

$\frac{\sin^{2} (5 {(lim_{x \to 0} x)}^{2}) + \sin^{2} (3 lim_{x \to 0} x)}{x^{2}}$

Step 9

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

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Step 9.1

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

$\frac{\sin^{2} (5 \cdot 0^{2}) + \sin^{2} (3 lim_{x \to 0} x)}{x^{2}}$

Step 9.2

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

$\frac{\sin^{2} (5 \cdot 0^{2}) + \sin^{2} (3 \cdot 0)}{x^{2}}$

Step 10

Simplify the answer.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Raising $0$ to any positive power yields $0$ .

$\frac{\sin^{2} (5 \cdot 0) + \sin^{2} (3 \cdot 0)}{x^{2}}$

Step 10.1.2

Multiply $5$ by $0$ .

$\frac{\sin^{2} (0) + \sin^{2} (3 \cdot 0)}{x^{2}}$

Step 10.1.3

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

$\frac{0^{2} + \sin^{2} (3 \cdot 0)}{x^{2}}$

Step 10.1.4

Raising $0$ to any positive power yields $0$ .

$\frac{0 + \sin^{2} (3 \cdot 0)}{x^{2}}$

Step 10.1.5

Multiply $3$ by $0$ .

$\frac{0 + \sin^{2} (0)}{x^{2}}$

Step 10.1.6

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

$\frac{0 + 0^{2}}{x^{2}}$

Step 10.1.7

Raising $0$ to any positive power yields $0$ .

$\frac{0 + 0}{x^{2}}$

Step 10.1.8

Add $0$ and $0$ .

$\frac{0}{x^{2}}$

Step 10.2

Divide $0$ by $x^{2}$ .

$0$