Calculus Examples

$\frac{lim_{x \to 5} 2 x^{2} + 15 x + 25}{5 - 4 x - x^{2}}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $5$ .

$\frac{lim_{x \to 5} 2 x^{2} + lim_{x \to 5} 15 x + lim_{x \to 5} 25}{5 - 4 x - x^{2}}$

Step 2

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

$\frac{2 lim_{x \to 5} x^{2} + lim_{x \to 5} 15 x + lim_{x \to 5} 25}{5 - 4 x - x^{2}}$

Step 3

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

$\frac{2 {(lim_{x \to 5} x)}^{2} + lim_{x \to 5} 15 x + lim_{x \to 5} 25}{5 - 4 x - x^{2}}$

Step 4

Move the term $15$ outside of the limit because it is constant with respect to $x$ .

$\frac{2 {(lim_{x \to 5} x)}^{2} + 15 lim_{x \to 5} x + lim_{x \to 5} 25}{5 - 4 x - x^{2}}$

Step 5

Evaluate the limit of $25$ which is constant as $x$ approaches $5$ .

$\frac{2 {(lim_{x \to 5} x)}^{2} + 15 lim_{x \to 5} x + 25}{5 - 4 x - x^{2}}$

Step 6

Evaluate the limits by plugging in $5$ for all occurrences of $x$ .

Tap for more steps...

Step 6.1

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$\frac{2 \cdot 5^{2} + 15 lim_{x \to 5} x + 25}{5 - 4 x - x^{2}}$

Step 6.2

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$\frac{2 \cdot 5^{2} + 15 \cdot 5 + 25}{5 - 4 x - x^{2}}$

Step 7

Simplify the answer.

Tap for more steps...

Step 7.1

Simplify the numerator.

Tap for more steps...

Step 7.1.1

Raise $5$ to the power of $2$ .

$\frac{2 \cdot 25 + 15 \cdot 5 + 25}{5 - 4 x - x^{2}}$

Step 7.1.2

Multiply $2$ by $25$ .

$\frac{50 + 15 \cdot 5 + 25}{5 - 4 x - x^{2}}$

Step 7.1.3

Multiply $15$ by $5$ .

$\frac{50 + 75 + 25}{5 - 4 x - x^{2}}$

Step 7.1.4

Add $50$ and $75$ .

$\frac{125 + 25}{5 - 4 x - x^{2}}$

Step 7.1.5

Add $125$ and $25$ .

$\frac{150}{5 - 4 x - x^{2}}$

Step 7.2

Simplify the denominator.

Tap for more steps...

Step 7.2.1

Rewrite $5$ as $9$ plus $- 4$

$\frac{150}{9 - 4 - 4 x - x^{2}}$

Step 7.2.2

Factor using the perfect square rule.

Tap for more steps...

Step 7.2.2.1

Rewrite $4$ as $2^{2}$ .

$\frac{150}{9 - (2^{2} + 4 x + x^{2})}$

Step 7.2.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 x = 2 \cdot 2 \cdot x$

Step 7.2.2.3

Rewrite the polynomial.

$\frac{150}{9 - (2^{2} + 2 \cdot 2 \cdot x + x^{2})}$

Step 7.2.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = 2$ and $b = x$ .

$\frac{150}{9 - {(2 + x)}^{2}}$

Step 7.2.3

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

$\frac{150}{3^{2} - {(2 + x)}^{2}}$

Step 7.2.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3$ and $b = 2 + x$ .

$\frac{150}{(3 + 2 + x) (3 - (2 + x))}$

Step 7.2.5

Simplify.

Tap for more steps...

Step 7.2.5.1

Add $3$ and $2$ .

$\frac{150}{(5 + x) (3 - (2 + x))}$

Step 7.2.5.2

Apply the distributive property.

$\frac{150}{(5 + x) (3 - 1 \cdot 2 - x)}$

Step 7.2.5.3

Multiply $- 1$ by $2$ .

$\frac{150}{(5 + x) (3 - 2 - x)}$

Step 7.2.5.4

Subtract $2$ from $3$ .

$\frac{150}{(5 + x) (1 - x)}$