Calculus Examples

$lim_{x \to \sqrt{3}} \frac{x^{2}}{\sqrt{12 - x^{2}}}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $\sqrt{3}$ .

$\frac{lim_{x \to \sqrt{3}} x^{2}}{lim_{x \to \sqrt{3}} \sqrt{12 - x^{2}}}$

Step 2

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

$\frac{{(lim_{x \to \sqrt{3}} x)}^{2}}{lim_{x \to \sqrt{3}} \sqrt{12 - x^{2}}}$

Step 3

Move the limit under the radical sign.

$\frac{{(lim_{x \to \sqrt{3}} x)}^{2}}{\sqrt{lim_{x \to \sqrt{3}} 12 - x^{2}}}$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\sqrt{3}$ .

$\frac{{(lim_{x \to \sqrt{3}} x)}^{2}}{\sqrt{lim_{x \to \sqrt{3}} 12 - lim_{x \to \sqrt{3}} x^{2}}}$

Step 5

Evaluate the limit of $12$ which is constant as $x$ approaches $\sqrt{3}$ .

$\frac{{(lim_{x \to \sqrt{3}} x)}^{2}}{\sqrt{12 - lim_{x \to \sqrt{3}} x^{2}}}$

Step 6

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

$\frac{{(lim_{x \to \sqrt{3}} x)}^{2}}{\sqrt{12 - {(lim_{x \to \sqrt{3}} x)}^{2}}}$

Step 7

Evaluate the limits by plugging in $\sqrt{3}$ for all occurrences of $x$ .

Tap for more steps...

Step 7.1

Evaluate the limit of $x$ by plugging in $\sqrt{3}$ for $x$ .

$\frac{{\sqrt{3}}^{2}}{\sqrt{12 - {(lim_{x \to \sqrt{3}} x)}^{2}}}$

Step 7.2

Evaluate the limit of $x$ by plugging in $\sqrt{3}$ for $x$ .

$\frac{{\sqrt{3}}^{2}}{\sqrt{12 - {\sqrt{3}}^{2}}}$

Step 8

Simplify the answer.

Tap for more steps...

Step 8.1

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Tap for more steps...

Step 8.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{{(3^{\frac{1}{2}})}^{2}}{\sqrt{12 - {\sqrt{3}}^{2}}}$

Step 8.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{3^{\frac{1}{2} \cdot 2}}{\sqrt{12 - {\sqrt{3}}^{2}}}$

Step 8.1.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{3^{\frac{2}{2}}}{\sqrt{12 - {\sqrt{3}}^{2}}}$

Step 8.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 8.1.4.1

Cancel the common factor.

$\frac{3^{\frac{2}{2}}}{\sqrt{12 - {\sqrt{3}}^{2}}}$

Step 8.1.4.2

Rewrite the expression.

$\frac{3^{1}}{\sqrt{12 - {\sqrt{3}}^{2}}}$

Step 8.1.5

Evaluate the exponent.

$\frac{3}{\sqrt{12 - {\sqrt{3}}^{2}}}$

Step 8.2

Simplify the denominator.

Tap for more steps...

Step 8.2.1

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Tap for more steps...

Step 8.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{3}{\sqrt{12 - {(3^{\frac{1}{2}})}^{2}}}$

Step 8.2.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{3}{\sqrt{12 - 3^{\frac{1}{2} \cdot 2}}}$

Step 8.2.1.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{3}{\sqrt{12 - 3^{\frac{2}{2}}}}$

Step 8.2.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 8.2.1.4.1

Cancel the common factor.

$\frac{3}{\sqrt{12 - 3^{\frac{2}{2}}}}$

Step 8.2.1.4.2

Rewrite the expression.

$\frac{3}{\sqrt{12 - 3^{1}}}$

Step 8.2.1.5

Evaluate the exponent.

$\frac{3}{\sqrt{12 - 1 \cdot 3}}$

Step 8.2.2

Multiply $- 1$ by $3$ .

$\frac{3}{\sqrt{12 - 3}}$

Step 8.2.3

Subtract $3$ from $12$ .

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

Step 8.2.4

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

$\frac{3}{\sqrt{3^{2}}}$

Step 8.2.5

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

$\frac{3}{3}$

Step 8.3

Divide $3$ by $3$ .

$1$