Calculus Examples

$\frac{lim_{x \to 8} 4 \sin (x)}{x^{2} + 2 x + 1}$

Step 1

Evaluate the limit.

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Step 1.1

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{4 lim_{x \to 8} \sin (x)}{x^{2} + 2 x + 1}$

Step 1.2

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

$\frac{4 \sin (lim_{x \to 8} x)}{x^{2} + 2 x + 1}$

Step 2

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

$\frac{4 \sin (8)}{x^{2} + 2 x + 1}$

Step 3

Simplify the answer.

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Step 3.1

Evaluate $\sin (8)$ .

$\frac{4 \cdot 0.1391731}{x^{2} + 2 x + 1}$

Step 3.2

Factor using the perfect square rule.

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Step 3.2.1

Rewrite $1$ as $1^{2}$ .

$\frac{4 \cdot 0.1391731}{x^{2} + 2 x + 1^{2}}$

Step 3.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Step 3.2.3

Rewrite the polynomial.

$\frac{4 \cdot 0.1391731}{x^{2} + 2 \cdot x \cdot 1 + 1^{2}}$

Step 3.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 1$ .

$\frac{4 \cdot 0.1391731}{{(x + 1)}^{2}}$

Step 3.3

Multiply $4$ by $0.1391731$ .

$\frac{0.5566924}{{(x + 1)}^{2}}$