Calculus Examples

$\frac{lim_{x \to 0} 3 - 3 \cos^{2} (x)}{21 x^{2}}$

Step 1

Evaluate the limit.

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Step 1.1

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

$\frac{lim_{x \to 0} 3 - lim_{x \to 0} 3 \cos^{2} (x)}{21 x^{2}}$

Step 1.2

Evaluate the limit of $3$ which is constant as $x$ approaches $0$ .

$\frac{3 - lim_{x \to 0} 3 \cos^{2} (x)}{21 x^{2}}$

Step 1.3

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

$\frac{3 - 3 lim_{x \to 0} \cos^{2} (x)}{21 x^{2}}$

Step 1.4

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

$\frac{3 - 3 {(lim_{x \to 0} \cos (x))}^{2}}{21 x^{2}}$

Step 1.5

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

$\frac{3 - 3 \cos^{2} (lim_{x \to 0} x)}{21 x^{2}}$

Step 2

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

$\frac{3 - 3 \cos^{2} (0)}{21 x^{2}}$

Step 3

Simplify the answer.

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Step 3.1

Factor $3$ out of $3$ .

$\frac{3 (1) - 3 \cos^{2} (0)}{21 x^{2}}$

Step 3.2

Factor $3$ out of $- 3 \cos^{2} (0)$ .

$\frac{3 (1) + 3 (- \cos^{2} (0))}{21 x^{2}}$

Step 3.3

Factor $3$ out of $3 (1) + 3 (- \cos^{2} (0))$ .

$\frac{3 (1 - \cos^{2} (0))}{21 x^{2}}$

Step 3.4

Apply pythagorean identity.

$\frac{3 \sin^{2} (0)}{21 x^{2}}$

Step 3.5

Cancel the common factor of $3$ and $21$ .

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Step 3.5.1

Factor $3$ out of $3 \sin^{2} (0)$ .

$\frac{3 (\sin^{2} (0))}{21 x^{2}}$

Step 3.5.2

Cancel the common factors.

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Step 3.5.2.1

Factor $3$ out of $21 x^{2}$ .

$\frac{3 (\sin^{2} (0))}{3 (7 x^{2})}$

Step 3.5.2.2

Cancel the common factor.

$\frac{3 \sin^{2} (0)}{3 (7 x^{2})}$

Step 3.5.2.3

Rewrite the expression.

$\frac{\sin^{2} (0)}{7 x^{2}}$

Step 3.6

Simplify the numerator.

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Step 3.6.1

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

$\frac{0^{2}}{7 x^{2}}$

Step 3.6.2

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

$\frac{0}{7 x^{2}}$

Step 3.7

Cancel the common factor of $0$ and $7$ .

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Step 3.7.1

Factor $7$ out of $0$ .

$\frac{7 (0)}{7 x^{2}}$

Step 3.7.2

Cancel the common factors.

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Step 3.7.2.1

Factor $7$ out of $7 x^{2}$ .

$\frac{7 (0)}{7 (x^{2})}$

Step 3.7.2.2

Cancel the common factor.

$\frac{7 \cdot 0}{7 x^{2}}$

Step 3.7.2.3

Rewrite the expression.

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

Step 3.8

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

$0$