Calculus Examples

$lim_{x \to 8} \frac{x + 2}{\sqrt{16 x^{2} + 1}}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $8$ .

$\frac{lim_{x \to 8} x + 2}{lim_{x \to 8} \sqrt{16 x^{2} + 1}}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{lim_{x \to 8} x + lim_{x \to 8} 2}{lim_{x \to 8} \sqrt{16 x^{2} + 1}}$

Step 3

Evaluate the limit of $2$ which is constant as $x$ approaches $8$ .

$\frac{lim_{x \to 8} x + 2}{lim_{x \to 8} \sqrt{16 x^{2} + 1}}$

Step 4

Move the limit under the radical sign.

$\frac{lim_{x \to 8} x + 2}{\sqrt{lim_{x \to 8} 16 x^{2} + 1}}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{lim_{x \to 8} x + 2}{\sqrt{lim_{x \to 8} 16 x^{2} + lim_{x \to 8} 1}}$

Step 6

Move the term $16$ outside of the limit because it is constant with respect to $x$ .

$\frac{lim_{x \to 8} x + 2}{\sqrt{16 lim_{x \to 8} x^{2} + lim_{x \to 8} 1}}$

Step 7

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

$\frac{lim_{x \to 8} x + 2}{\sqrt{16 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} 1}}$

Step 8

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

$\frac{lim_{x \to 8} x + 2}{\sqrt{16 {(lim_{x \to 8} x)}^{2} + 1}}$

Step 9

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

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Step 9.1

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

$\frac{8 + 2}{\sqrt{16 {(lim_{x \to 8} x)}^{2} + 1}}$

Step 9.2

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

$\frac{8 + 2}{\sqrt{16 \cdot 8^{2} + 1}}$

Step 10

Simplify the answer.

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Step 10.1

Add $8$ and $2$ .

$\frac{10}{\sqrt{16 \cdot 8^{2} + 1}}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

Raise $8$ to the power of $2$ .

$\frac{10}{\sqrt{16 \cdot 64 + 1}}$

Step 10.2.2

Multiply $16$ by $64$ .

$\frac{10}{\sqrt{1024 + 1}}$

Step 10.2.3

Add $1024$ and $1$ .

$\frac{10}{\sqrt{1025}}$

Step 10.2.4

Rewrite $1025$ as $5^{2} \cdot 41$ .

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Step 10.2.4.1

Factor $25$ out of $1025$ .

$\frac{10}{\sqrt{25 (41)}}$

Step 10.2.4.2

Rewrite $25$ as $5^{2}$ .

$\frac{10}{\sqrt{5^{2} \cdot 41}}$

Step 10.2.5

Pull terms out from under the radical.

$\frac{10}{5 \sqrt{41}}$

Step 10.3

Cancel the common factor of $10$ and $5$ .

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Step 10.3.1

Factor $5$ out of $10$ .

$\frac{5 \cdot 2}{5 \sqrt{41}}$

Step 10.3.2

Cancel the common factors.

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Step 10.3.2.1

Factor $5$ out of $5 \sqrt{41}$ .

$\frac{5 \cdot 2}{5 (\sqrt{41})}$

Step 10.3.2.2

Cancel the common factor.

$\frac{5 \cdot 2}{5 \sqrt{41}}$

Step 10.3.2.3

Rewrite the expression.

$\frac{2}{\sqrt{41}}$

Step 10.4

Multiply $\frac{2}{\sqrt{41}}$ by $\frac{\sqrt{41}}{\sqrt{41}}$ .

$\frac{2}{\sqrt{41}} \cdot \frac{\sqrt{41}}{\sqrt{41}}$

Step 10.5

Combine and simplify the denominator.

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Step 10.5.1

Multiply $\frac{2}{\sqrt{41}}$ by $\frac{\sqrt{41}}{\sqrt{41}}$ .

$\frac{2 \sqrt{41}}{\sqrt{41} \sqrt{41}}$

Step 10.5.2

Raise $\sqrt{41}$ to the power of $1$ .

$\frac{2 \sqrt{41}}{{\sqrt{41}}^{1} \sqrt{41}}$

Step 10.5.3

Raise $\sqrt{41}$ to the power of $1$ .

$\frac{2 \sqrt{41}}{{\sqrt{41}}^{1} {\sqrt{41}}^{1}}$

Step 10.5.4

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

$\frac{2 \sqrt{41}}{{\sqrt{41}}^{1 + 1}}$

Step 10.5.5

Add $1$ and $1$ .

$\frac{2 \sqrt{41}}{{\sqrt{41}}^{2}}$

Step 10.5.6

Rewrite ${\sqrt{41}}^{2}$ as $41$ .

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Step 10.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{41}$ as $41^{\frac{1}{2}}$ .

$\frac{2 \sqrt{41}}{{(41^{\frac{1}{2}})}^{2}}$

Step 10.5.6.2

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

$\frac{2 \sqrt{41}}{41^{\frac{1}{2} \cdot 2}}$

Step 10.5.6.3

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

$\frac{2 \sqrt{41}}{41^{\frac{2}{2}}}$

Step 10.5.6.4

Cancel the common factor of $2$ .

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Step 10.5.6.4.1

Cancel the common factor.

$\frac{2 \sqrt{41}}{41^{\frac{2}{2}}}$

Step 10.5.6.4.2

Rewrite the expression.

$\frac{2 \sqrt{41}}{41^{1}}$

Step 10.5.6.5

Evaluate the exponent.

$\frac{2 \sqrt{41}}{41}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt{41}}{41}$

Decimal Form:

$0.31234752 \dots$