Calculus Examples

$\frac{lim_{x \to 25} 2 x}{\sqrt{625 - x^{2}}}$

Step 1

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{2 lim_{x \to 25} x}{\sqrt{625 - x^{2}}}$

Step 2

Evaluate the limit of $x$ by plugging in $25$ for $x$ .

$\frac{2 \cdot 25}{\sqrt{625 - x^{2}}}$

Step 3

Simplify the answer.

Tap for more steps...

Step 3.1

Multiply $2$ by $25$ .

$\frac{50}{\sqrt{625 - x^{2}}}$

Step 3.2

Simplify the denominator.

Tap for more steps...

Step 3.2.1

Rewrite $625$ as $25^{2}$ .

$\frac{50}{\sqrt{25^{2} - x^{2}}}$

Step 3.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 25$ and $b = x$ .

$\frac{50}{\sqrt{(25 + x) (25 - x)}}$

Step 3.3

Multiply $\frac{50}{\sqrt{(25 + x) (25 - x)}}$ by $\frac{\sqrt{(25 + x) (25 - x)}}{\sqrt{(25 + x) (25 - x)}}$ .

$\frac{50}{\sqrt{(25 + x) (25 - x)}} \cdot \frac{\sqrt{(25 + x) (25 - x)}}{\sqrt{(25 + x) (25 - x)}}$

Step 3.4

Combine and simplify the denominator.

Tap for more steps...

Step 3.4.1

Multiply $\frac{50}{\sqrt{(25 + x) (25 - x)}}$ by $\frac{\sqrt{(25 + x) (25 - x)}}{\sqrt{(25 + x) (25 - x)}}$ .

$\frac{50 \sqrt{(25 + x) (25 - x)}}{\sqrt{(25 + x) (25 - x)} \sqrt{(25 + x) (25 - x)}}$

Step 3.4.2

Raise $\sqrt{(25 + x) (25 - x)}$ to the power of $1$ .

$\frac{50 \sqrt{(25 + x) (25 - x)}}{{\sqrt{(25 + x) (25 - x)}}^{1} \sqrt{(25 + x) (25 - x)}}$

Step 3.4.3

Raise $\sqrt{(25 + x) (25 - x)}$ to the power of $1$ .

$\frac{50 \sqrt{(25 + x) (25 - x)}}{{\sqrt{(25 + x) (25 - x)}}^{1} {\sqrt{(25 + x) (25 - x)}}^{1}}$

Step 3.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{50 \sqrt{(25 + x) (25 - x)}}{{\sqrt{(25 + x) (25 - x)}}^{1 + 1}}$

Step 3.4.5

Add $1$ and $1$ .

$\frac{50 \sqrt{(25 + x) (25 - x)}}{{\sqrt{(25 + x) (25 - x)}}^{2}}$

Step 3.4.6

Rewrite ${\sqrt{(25 + x) (25 - x)}}^{2}$ as $(25 + x) (25 - x)$ .

Tap for more steps...

Step 3.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(25 + x) (25 - x)}$ as ${((25 + x) (25 - x))}^{\frac{1}{2}}$ .

$\frac{50 \sqrt{(25 + x) (25 - x)}}{{({((25 + x) (25 - x))}^{\frac{1}{2}})}^{2}}$

Step 3.4.6.2

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

$\frac{50 \sqrt{(25 + x) (25 - x)}}{{((25 + x) (25 - x))}^{\frac{1}{2} \cdot 2}}$

Step 3.4.6.3

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

$\frac{50 \sqrt{(25 + x) (25 - x)}}{{((25 + x) (25 - x))}^{\frac{2}{2}}}$

Step 3.4.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.4.6.4.1

Cancel the common factor.

$\frac{50 \sqrt{(25 + x) (25 - x)}}{{((25 + x) (25 - x))}^{\frac{2}{2}}}$

Step 3.4.6.4.2

Rewrite the expression.

$\frac{50 \sqrt{(25 + x) (25 - x)}}{{((25 + x) (25 - x))}^{1}}$

Step 3.4.6.5

Simplify.

$\frac{50 \sqrt{(25 + x) (25 - x)}}{(25 + x) (25 - x)}$