Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

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

Step 8

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

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

Step 9

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

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Step 9.1

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

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

Step 9.2

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

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

Step 9.3

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

$8^{2} + \sin (8) \cdot \sin (\frac{1}{8^{2}})$

Step 10

Simplify the answer.

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Step 10.1

Simplify each term.

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Step 10.1.1

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

$64 + \sin (8) \cdot \sin (\frac{1}{8^{2}})$

Step 10.1.2

Evaluate $\sin (8)$ .

$64 + 0.1391731 \cdot \sin (\frac{1}{8^{2}})$

Step 10.1.3

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

$64 + 0.1391731 \cdot \sin (\frac{1}{64})$

Step 10.1.4

Evaluate $\sin (\frac{1}{64})$ .

$64 + 0.1391731 \cdot 0.0002727$

Step 10.1.5

Multiply $0.1391731$ by $0.0002727$ .

$64 + 0.00003795$

Step 10.2

Add $64$ and $0.00003795$ .

$64.00003795$