Calculus Examples

$lim_{x \to 8} \frac{\sqrt{10 + 8 x^{2}}}{2 + 3 x}$

Step 1

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

$\frac{lim_{x \to 8} \sqrt{10 + 8 x^{2}}}{lim_{x \to 8} 2 + 3 x}$

Step 2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 8} 10 + 8 x^{2}}}{lim_{x \to 8} 2 + 3 x}$

Step 3

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

$\frac{\sqrt{lim_{x \to 8} 10 + lim_{x \to 8} 8 x^{2}}}{lim_{x \to 8} 2 + 3 x}$

Step 4

Evaluate the limit of $10$ which is constant as $x$ approaches $8$ .

$\frac{\sqrt{10 + lim_{x \to 8} 8 x^{2}}}{lim_{x \to 8} 2 + 3 x}$

Step 5

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sqrt{10 + 8 lim_{x \to 8} x^{2}}}{lim_{x \to 8} 2 + 3 x}$

Step 6

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

$\frac{\sqrt{10 + 8 {(lim_{x \to 8} x)}^{2}}}{lim_{x \to 8} 2 + 3 x}$

Step 7

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

$\frac{\sqrt{10 + 8 {(lim_{x \to 8} x)}^{2}}}{lim_{x \to 8} 2 + lim_{x \to 8} 3 x}$

Step 8

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

$\frac{\sqrt{10 + 8 {(lim_{x \to 8} x)}^{2}}}{2 + lim_{x \to 8} 3 x}$

Step 9

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

$\frac{\sqrt{10 + 8 {(lim_{x \to 8} x)}^{2}}}{2 + 3 lim_{x \to 8} x}$

Step 10

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

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Step 10.1

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

$\frac{\sqrt{10 + 8 \cdot 8^{2}}}{2 + 3 lim_{x \to 8} x}$

Step 10.2

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

$\frac{\sqrt{10 + 8 \cdot 8^{2}}}{2 + 3 \cdot 8}$

Step 11

Simplify the answer.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Multiply $8$ by $8^{2}$ by adding the exponents.

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Step 11.1.1.1

Multiply $8$ by $8^{2}$ .

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Step 11.1.1.1.1

Raise $8$ to the power of $1$ .

$\frac{\sqrt{10 + 8^{1} \cdot 8^{2}}}{2 + 3 \cdot 8}$

Step 11.1.1.1.2

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

$\frac{\sqrt{10 + 8^{1 + 2}}}{2 + 3 \cdot 8}$

Step 11.1.1.2

Add $1$ and $2$ .

$\frac{\sqrt{10 + 8^{3}}}{2 + 3 \cdot 8}$

Step 11.1.2

Raise $8$ to the power of $3$ .

$\frac{\sqrt{10 + 512}}{2 + 3 \cdot 8}$

Step 11.1.3

Add $10$ and $512$ .

$\frac{\sqrt{522}}{2 + 3 \cdot 8}$

Step 11.1.4

Rewrite $522$ as $3^{2} \cdot 58$ .

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Step 11.1.4.1

Factor $9$ out of $522$ .

$\frac{\sqrt{9 (58)}}{2 + 3 \cdot 8}$

Step 11.1.4.2

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{3^{2} \cdot 58}}{2 + 3 \cdot 8}$

Step 11.1.5

Pull terms out from under the radical.

$\frac{3 \sqrt{58}}{2 + 3 \cdot 8}$

Step 11.2

Simplify the denominator.

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Step 11.2.1

Multiply $3$ by $8$ .

$\frac{3 \sqrt{58}}{2 + 24}$

Step 11.2.2

Add $2$ and $24$ .

$\frac{3 \sqrt{58}}{26}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{3 \sqrt{58}}{26}$

Decimal Form:

$0.87874305 \dots$