Calculus Examples

$\frac{lim_{x \to 5} x - 5}{x^{2} - 10 x + 25}$

Step 1

Evaluate the limit.

Tap for more steps...

Step 1.1

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

$\frac{lim_{x \to 5} x - lim_{x \to 5} 5}{x^{2} - 10 x + 25}$

Step 1.2

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

$\frac{lim_{x \to 5} x - 1 \cdot 5}{x^{2} - 10 x + 25}$

Step 2

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

$\frac{5 - 1 \cdot 5}{x^{2} - 10 x + 25}$

Step 3

Simplify the answer.

Tap for more steps...

Step 3.1

Simplify the numerator.

Tap for more steps...

Step 3.1.1

Multiply $- 1$ by $5$ .

$\frac{5 - 5}{x^{2} - 10 x + 25}$

Step 3.1.2

Subtract $5$ from $5$ .

$\frac{0}{x^{2} - 10 x + 25}$

Step 3.2

Factor using the perfect square rule.

Tap for more steps...

Step 3.2.1

Rewrite $25$ as $5^{2}$ .

$\frac{0}{x^{2} - 10 x + 5^{2}}$

Step 3.2.2

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

$10 x = 2 \cdot x \cdot 5$

Step 3.2.3

Rewrite the polynomial.

$\frac{0}{x^{2} - 2 \cdot x \cdot 5 + 5^{2}}$

Step 3.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 5$ .

$\frac{0}{{(x - 5)}^{2}}$

Step 3.3

Divide $0$ by ${(x - 5)}^{2}$ .

$0$