Calculus Examples

$lim_{x \to 8} \frac{6 \sqrt{1 + x^{2}}}{3 x^{2}}$

Step 1

Cancel the common factor of $6$ and $3$ .

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Step 1.1

Factor $3$ out of $6 \sqrt{1 + x^{2}}$ .

$lim_{x \to 8} \frac{3 (2 \sqrt{1 + x^{2}})}{3 x^{2}}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $3$ out of $3 x^{2}$ .

$lim_{x \to 8} \frac{3 (2 \sqrt{1 + x^{2}})}{3 (x^{2})}$

Step 1.2.2

Cancel the common factor.

$lim_{x \to 8} \frac{3 (2 \sqrt{1 + x^{2}})}{3 x^{2}}$

Step 1.2.3

Rewrite the expression.

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

Step 2

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

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

Step 3

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

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

Step 4

Move the limit under the radical sign.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

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 $x$ .

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

Step 9.2

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

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

Step 10

Simplify the answer.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

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

$2 \frac{\sqrt{1 + 64}}{8^{2}}$

Step 10.1.2

Add $1$ and $64$ .

$2 \frac{\sqrt{65}}{8^{2}}$

Step 10.2

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

$2 \frac{\sqrt{65}}{64}$

Step 10.3

Cancel the common factor of $2$ .

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Step 10.3.1

Factor $2$ out of $64$ .

$2 \frac{\sqrt{65}}{2 (32)}$

Step 10.3.2

Cancel the common factor.

$2 \frac{\sqrt{65}}{2 \cdot 32}$

Step 10.3.3

Rewrite the expression.

$\frac{\sqrt{65}}{32}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{65}}{32}$

Decimal Form:

$0.25194555 \dots$