Calculus Examples

$lim_{x \to 303} \frac{x - 303}{\sqrt{x + 21 - 18}}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $303$ .

$\frac{lim_{x \to 303} x - 303}{lim_{x \to 303} \sqrt{x + 21 - 18}}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $303$ .

$\frac{lim_{x \to 303} x - lim_{x \to 303} 303}{lim_{x \to 303} \sqrt{x + 21 - 18}}$

Step 3

Evaluate the limit of $303$ which is constant as $x$ approaches $303$ .

$\frac{lim_{x \to 303} x - 1 \cdot 303}{lim_{x \to 303} \sqrt{x + 21 - 18}}$

Step 4

Move the limit under the radical sign.

$\frac{lim_{x \to 303} x - 1 \cdot 303}{\sqrt{lim_{x \to 303} x + 21 - 18}}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $303$ .

$\frac{lim_{x \to 303} x - 1 \cdot 303}{\sqrt{lim_{x \to 303} x + lim_{x \to 303} 21 - lim_{x \to 303} 18}}$

Step 6

Evaluate the limit of $21$ which is constant as $x$ approaches $303$ .

$\frac{lim_{x \to 303} x - 1 \cdot 303}{\sqrt{lim_{x \to 303} x + 21 - lim_{x \to 303} 18}}$

Step 7

Evaluate the limit of $18$ which is constant as $x$ approaches $303$ .

$\frac{lim_{x \to 303} x - 1 \cdot 303}{\sqrt{lim_{x \to 303} x + 21 - 1 \cdot 18}}$

Step 8

Evaluate the limits by plugging in $303$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $x$ by plugging in $303$ for $x$ .

$\frac{303 - 1 \cdot 303}{\sqrt{lim_{x \to 303} x + 21 - 1 \cdot 18}}$

Step 8.2

Evaluate the limit of $x$ by plugging in $303$ for $x$ .

$\frac{303 - 1 \cdot 303}{\sqrt{303 + 21 - 1 \cdot 18}}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Multiply $- 1$ by $303$ .

$\frac{303 - 303}{\sqrt{303 + 21 - 1 \cdot 18}}$

Step 9.1.2

Subtract $303$ from $303$ .

$\frac{0}{\sqrt{303 + 21 - 1 \cdot 18}}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

Multiply $- 1$ by $18$ .

$\frac{0}{\sqrt{303 + 21 - 18}}$

Step 9.2.2

Add $303$ and $21$ .

$\frac{0}{\sqrt{324 - 18}}$

Step 9.2.3

Subtract $18$ from $324$ .

$\frac{0}{\sqrt{306}}$

Step 9.2.4

Rewrite $306$ as $3^{2} \cdot 34$ .

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Step 9.2.4.1

Factor $9$ out of $306$ .

$\frac{0}{\sqrt{9 (34)}}$

Step 9.2.4.2

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

$\frac{0}{\sqrt{3^{2} \cdot 34}}$

Step 9.2.5

Pull terms out from under the radical.

$\frac{0}{3 \sqrt{34}}$

Step 9.3

Cancel the common factor of $0$ and $3$ .

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Step 9.3.1

Factor $3$ out of $0$ .

$\frac{3 (0)}{3 \sqrt{34}}$

Step 9.3.2

Cancel the common factors.

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Step 9.3.2.1

Factor $3$ out of $3 \sqrt{34}$ .

$\frac{3 (0)}{3 (\sqrt{34})}$

Step 9.3.2.2

Cancel the common factor.

$\frac{3 \cdot 0}{3 \sqrt{34}}$

Step 9.3.2.3

Rewrite the expression.

$\frac{0}{\sqrt{34}}$

Step 9.4

Divide $0$ by $\sqrt{34}$ .

$0$