Calculus Examples

$lim_{x \to 0} \arctan (\frac{1 - x}{\sqrt{3 + x}})$

Step 1

Simplify the answer.

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Step 1.1

Add $1$ and $0$ .

$\frac{1}{\sqrt{3 + 0}}$

Step 1.2

Add $3$ and $0$ .

$\frac{1}{\sqrt{3}}$

Step 1.3

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 1.4

Combine and simplify the denominator.

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Step 1.4.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 1.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 1.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 1.4.5

Add $1$ and $1$ .

$\frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

Step 1.4.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 1.4.6.2

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

$\frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 1.4.6.3

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

$\frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 1.4.6.4

Cancel the common factor of $2$ .

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Step 1.4.6.4.1

Cancel the common factor.

$\frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 1.4.6.4.2

Rewrite the expression.

$\frac{\sqrt{3}}{3^{1}}$

Step 1.4.6.5

Evaluate the exponent.

$\frac{\sqrt{3}}{3}$

Step 2

Substitute $t$ for $\frac{1 - x}{\sqrt{3 + x}}$ and let $t$ approach $\frac{\sqrt{3}}{3}$ since $lim_{x \to 0} \frac{1 - x}{\sqrt{3 + x}} = \frac{\sqrt{3}}{3}$ .

$lim_{t \to \frac{\sqrt{3}}{3}} \arctan (t)$

Step 3

Evaluate the limits by plugging in $\frac{\sqrt{3}}{3}$ for all occurrences of $t$ .

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Step 3.1

Simplify the answer.

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Step 3.1.1

Add $1$ and $0$ .

$\arctan (\frac{1}{\sqrt{3 + 0}})$

Step 3.1.2

Add $3$ and $0$ .

$\arctan (\frac{1}{\sqrt{3}})$

Step 3.1.3

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\arctan (\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 3.1.4

Combine and simplify the denominator.

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Step 3.1.4.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\arctan (\frac{\sqrt{3}}{\sqrt{3} \sqrt{3}})$

Step 3.1.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$\arctan (\frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}})$

Step 3.1.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$\arctan (\frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}})$

Step 3.1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\arctan (\frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}})$

Step 3.1.4.5

Add $1$ and $1$ .

$\arctan (\frac{\sqrt{3}}{{\sqrt{3}}^{2}})$

Step 3.1.4.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.1.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\arctan (\frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}})$

Step 3.1.4.6.2

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

$\arctan (\frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}})$

Step 3.1.4.6.3

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

$\arctan (\frac{\sqrt{3}}{3^{\frac{2}{2}}})$

Step 3.1.4.6.4

Cancel the common factor of $2$ .

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Step 3.1.4.6.4.1

Cancel the common factor.

$\arctan (\frac{\sqrt{3}}{3^{\frac{2}{2}}})$

Step 3.1.4.6.4.2

Rewrite the expression.

$\arctan (\frac{\sqrt{3}}{3^{1}})$

Step 3.1.4.6.5

Evaluate the exponent.

$\arctan (\frac{\sqrt{3}}{3})$

Step 3.2

The exact value of $\arctan (\frac{\sqrt{3}}{3})$ is $\frac{π}{6}$ .

$\frac{π}{6}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{π}{6}$

Decimal Form:

$0.52359877 \dots$