Calculus Examples

$lim_{x \to 9} \frac{40 + \sqrt{x}}{\sqrt{40 + x}}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $9$ .

$\frac{lim_{x \to 9} 40 + \sqrt{x}}{lim_{x \to 9} \sqrt{40 + x}}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $9$ .

$\frac{lim_{x \to 9} 40 + lim_{x \to 9} \sqrt{x}}{lim_{x \to 9} \sqrt{40 + x}}$

Step 3

Evaluate the limit of $40$ which is constant as $x$ approaches $9$ .

$\frac{40 + lim_{x \to 9} \sqrt{x}}{lim_{x \to 9} \sqrt{40 + x}}$

Step 4

Move the limit under the radical sign.

$\frac{40 + \sqrt{lim_{x \to 9} x}}{lim_{x \to 9} \sqrt{40 + x}}$

Step 5

Move the limit under the radical sign.

$\frac{40 + \sqrt{lim_{x \to 9} x}}{\sqrt{lim_{x \to 9} 40 + x}}$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $9$ .

$\frac{40 + \sqrt{lim_{x \to 9} x}}{\sqrt{lim_{x \to 9} 40 + lim_{x \to 9} x}}$

Step 7

Evaluate the limit of $40$ which is constant as $x$ approaches $9$ .

$\frac{40 + \sqrt{lim_{x \to 9} x}}{\sqrt{40 + lim_{x \to 9} x}}$

Step 8

Evaluate the limits by plugging in $9$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $x$ by plugging in $9$ for $x$ .

$\frac{40 + \sqrt{9}}{\sqrt{40 + lim_{x \to 9} x}}$

Step 8.2

Evaluate the limit of $x$ by plugging in $9$ for $x$ .

$\frac{40 + \sqrt{9}}{\sqrt{40 + 9}}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

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

$\frac{40 + \sqrt{3^{2}}}{\sqrt{40 + 9}}$

Step 9.1.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{40 + 3}{\sqrt{40 + 9}}$

Step 9.1.3

Add $40$ and $3$ .

$\frac{43}{\sqrt{40 + 9}}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

Add $40$ and $9$ .

$\frac{43}{\sqrt{49}}$

Step 9.2.2

Rewrite $49$ as $7^{2}$ .

$\frac{43}{\sqrt{7^{2}}}$

Step 9.2.3

Pull terms out from under the radical, assuming positive real numbers.

$\frac{43}{7}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{43}{7}$

Decimal Form:

$6.\overline{142857}$