Calculus Examples

$lim_{x \to 8} \frac{\sin (x^{2})}{\sin (x)}$

Step 1

Apply trigonometric identities.

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Step 1.1

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

$lim_{x \to 8} \sin (x^{2}) \frac{1}{\sin (x)}$

Step 1.2

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

$lim_{x \to 8} \frac{\sin (x^{2})}{1} \cdot \frac{1}{\sin (x)}$

Step 1.3

Simplify.

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Step 1.3.1

Divide $\sin (x^{2})$ by $1$ .

$lim_{x \to 8} \sin (x^{2}) \frac{1}{\sin (x)}$

Step 1.3.2

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$lim_{x \to 8} \sin (x^{2}) \csc (x)$

Step 2

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $8$ .

$lim_{x \to 8} \sin (x^{2}) \cdot lim_{x \to 8} \csc (x)$

Step 3

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

$\sin (lim_{x \to 8} x^{2}) \cdot lim_{x \to 8} \csc (x)$

Step 4

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

$\sin ({(lim_{x \to 8} x)}^{2}) \cdot lim_{x \to 8} \csc (x)$

Step 5

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

$\sin ({(lim_{x \to 8} x)}^{2}) \cdot \csc (lim_{x \to 8} x)$

Step 6

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

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Step 6.1

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\sin (8^{2}) \cdot \csc (lim_{x \to 8} x)$

Step 6.2

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\sin (8^{2}) \cdot \csc (8)$

Step 7

Simplify the answer.

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Step 7.1

Raise $8$ to the power of $2$ .

$\sin (64) \cdot \csc (8)$

Step 7.2

Evaluate $\sin (64)$ .

$0.89879404 \cdot \csc (8)$

Step 7.3

Evaluate $\csc (8)$ .

$0.89879404 \cdot 7.18529653$

Step 7.4

Multiply $0.89879404$ by $7.18529653$ .

$6.45810174$