Calculus Examples

$lim_{x \to 7} \frac{\sqrt{x + 7}}{x^{2} + 1}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $7$ .

$\frac{lim_{x \to 7} \sqrt{x + 7}}{lim_{x \to 7} x^{2} + 1}$

Step 2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 7} x + 7}}{lim_{x \to 7} x^{2} + 1}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $7$ .

$\frac{\sqrt{lim_{x \to 7} x + lim_{x \to 7} 7}}{lim_{x \to 7} x^{2} + 1}$

Step 4

Evaluate the limit of $7$ which is constant as $x$ approaches $7$ .

$\frac{\sqrt{lim_{x \to 7} x + 7}}{lim_{x \to 7} x^{2} + 1}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $7$ .

$\frac{\sqrt{lim_{x \to 7} x + 7}}{lim_{x \to 7} x^{2} + lim_{x \to 7} 1}$

Step 6

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

$\frac{\sqrt{lim_{x \to 7} x + 7}}{{(lim_{x \to 7} x)}^{2} + lim_{x \to 7} 1}$

Step 7

Evaluate the limit of $1$ which is constant as $x$ approaches $7$ .

$\frac{\sqrt{lim_{x \to 7} x + 7}}{{(lim_{x \to 7} x)}^{2} + 1}$

Step 8

Evaluate the limits by plugging in $7$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $x$ by plugging in $7$ for $x$ .

$\frac{\sqrt{7 + 7}}{{(lim_{x \to 7} x)}^{2} + 1}$

Step 8.2

Evaluate the limit of $x$ by plugging in $7$ for $x$ .

$\frac{\sqrt{7 + 7}}{7^{2} + 1}$

Step 9

Simplify the answer.

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Step 9.1

Add $7$ and $7$ .

$\frac{\sqrt{14}}{7^{2} + 1}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

Raise $7$ to the power of $2$ .

$\frac{\sqrt{14}}{49 + 1}$

Step 9.2.2

Add $49$ and $1$ .

$\frac{\sqrt{14}}{50}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{14}}{50}$

Decimal Form:

$0.07483314 \dots$