Calculus Examples

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

Step 1

Divide $\sqrt{x^{2} + 1} - x$ by $1$ .

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

Step 2

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

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

Step 3

Move the limit under the radical sign.

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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Step 7.1

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

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

Step 7.2

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

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

Step 8

Simplify each term.

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Step 8.1

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

$\sqrt{64 + 1} - 8$

Step 8.2

Add $64$ and $1$ .

$\sqrt{65} - 8$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\sqrt{65} - 8$

Decimal Form:

$0.06225774 \dots$