Calculus Examples

$lim_{x \to \frac{π}{4}} \sin^{2} (x)$

Step 1

Evaluate the limit.

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Step 1.1

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

${(lim_{x \to \frac{π}{4}} \sin (x))}^{2}$

Step 1.2

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

$\sin^{2} (lim_{x \to \frac{π}{4}} x)$

Step 2

Evaluate the limit of $x$ by plugging in $\frac{π}{4}$ for $x$ .

$\sin^{2} (\frac{π}{4})$

Step 3

Simplify the answer.

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Step 3.1

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

${(\frac{\sqrt{2}}{2})}^{2}$

Step 3.2

Apply the product rule to $\frac{\sqrt{2}}{2}$ .

$\frac{{\sqrt{2}}^{2}}{2^{2}}$

Step 3.3

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{{(2^{\frac{1}{2}})}^{2}}{2^{2}}$

Step 3.3.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2^{\frac{1}{2} \cdot 2}}{2^{2}}$

Step 3.3.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{2^{\frac{2}{2}}}{2^{2}}$

Step 3.3.4

Cancel the common factor of $2$ .

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Step 3.3.4.1

Cancel the common factor.

$\frac{2^{\frac{2}{2}}}{2^{2}}$

Step 3.3.4.2

Rewrite the expression.

$\frac{2^{1}}{2^{2}}$

Step 3.3.5

Evaluate the exponent.

$\frac{2}{2^{2}}$

Step 3.4

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

$\frac{2}{4}$

Step 3.5

Cancel the common factor of $2$ and $4$ .

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Step 3.5.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{4}$

Step 3.5.2

Cancel the common factors.

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Step 3.5.2.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 1}{2 \cdot 2}$

Step 3.5.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 2}$

Step 3.5.2.3

Rewrite the expression.

$\frac{1}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$