Calculus Examples

$lim_{h \to 0} \frac{3}{\sqrt{3 h + 1 + 1}}$

Step 1

Evaluate the limit.

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Step 1.1

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

$3 lim_{h \to 0} \frac{1}{\sqrt{3 h + 1 + 1}}$

Step 1.2

Split the limit using the Limits Quotient Rule on the limit as $h$ approaches $0$ .

$3 \frac{lim_{h \to 0} 1}{lim_{h \to 0} \sqrt{3 h + 1 + 1}}$

Step 1.3

Evaluate the limit of $1$ which is constant as $h$ approaches $0$ .

$3 \frac{1}{lim_{h \to 0} \sqrt{3 h + 1 + 1}}$

Step 1.4

Move the limit under the radical sign.

$3 \frac{1}{\sqrt{lim_{h \to 0} 3 h + 1 + 1}}$

Step 1.5

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

$3 \frac{1}{\sqrt{lim_{h \to 0} 3 h + lim_{h \to 0} 1 + lim_{h \to 0} 1}}$

Step 1.6

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

$3 \frac{1}{\sqrt{3 lim_{h \to 0} h + lim_{h \to 0} 1 + lim_{h \to 0} 1}}$

Step 1.7

Evaluate the limit of $1$ which is constant as $h$ approaches $0$ .

$3 \frac{1}{\sqrt{3 lim_{h \to 0} h + 1 + lim_{h \to 0} 1}}$

Step 1.8

Evaluate the limit of $1$ which is constant as $h$ approaches $0$ .

$3 \frac{1}{\sqrt{3 lim_{h \to 0} h + 1 + 1}}$

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

Simplify the denominator.

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Step 3.1.1

Multiply $3$ by $0$ .

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

Step 3.1.2

Add $0$ and $1$ .

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

Step 3.1.3

Add $1$ and $1$ .

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

Step 3.2

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 3.3

Combine and simplify the denominator.

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Step 3.3.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 3.3.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 3.3.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 3.3.4

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

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

Step 3.3.5

Add $1$ and $1$ .

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

Step 3.3.6

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

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Step 3.3.6.1

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

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

Step 3.3.6.2

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

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

Step 3.3.6.3

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

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

Step 3.3.6.4

Cancel the common factor of $2$ .

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Step 3.3.6.4.1

Cancel the common factor.

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

Step 3.3.6.4.2

Rewrite the expression.

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

Step 3.3.6.5

Evaluate the exponent.

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

Step 3.4

Combine $3$ and $\frac{\sqrt{2}}{2}$ .

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$2.12132034 \dots$