Calculus Examples

\frac{lim x \to 5 x^{2} - 3 x - 10}{x^{2} - 10 x + 25}

$\frac{lim_{x \to 5} x^{2} - 3 x - 10}{x^{2} - 10 x + 25}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as

x

$x$ approaches

5

$5$ .

\frac{lim x \to 5 x^{2} - lim x \to 5 3 x - lim x \to 5 10}{x^{2} - 10 x + 25}

$\frac{lim_{x \to 5} x^{2} - lim_{x \to 5} 3 x - lim_{x \to 5} 10}{x^{2} - 10 x + 25}$

Step 2

Move the exponent

2

$2$ from

x^{2}

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

\frac{{(lim x \to 5 x)}^{2} - lim x \to 5 3 x - lim x \to 5 10}{x^{2} - 10 x + 25}

$\frac{{(lim_{x \to 5} x)}^{2} - lim_{x \to 5} 3 x - lim_{x \to 5} 10}{x^{2} - 10 x + 25}$

Step 3

Move the term

3

$3$ outside of the limit because it is constant with respect to

x

$x$ .

\frac{{(lim x \to 5 x)}^{2} - 3 lim x \to 5 x - lim x \to 5 10}{x^{2} - 10 x + 25}

$\frac{{(lim_{x \to 5} x)}^{2} - 3 lim_{x \to 5} x - lim_{x \to 5} 10}{x^{2} - 10 x + 25}$

Step 4

Evaluate the limit of

10

$10$ which is constant as

x

$x$ approaches

5

$5$ .

\frac{{(lim x \to 5 x)}^{2} - 3 lim x \to 5 x - 1 \cdot 10}{x^{2} - 10 x + 25}

$\frac{{(lim_{x \to 5} x)}^{2} - 3 lim_{x \to 5} x - 1 \cdot 10}{x^{2} - 10 x + 25}$

Step 5

Evaluate the limits by plugging in

5

$5$ for all occurrences of

x

$x$ .

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Step 5.1

Evaluate the limit of

x

$x$ by plugging in

5

$5$ for

x

$x$ .

\frac{5^{2} - 3 lim x \to 5 x - 1 \cdot 10}{x^{2} - 10 x + 25}

$\frac{5^{2} - 3 lim_{x \to 5} x - 1 \cdot 10}{x^{2} - 10 x + 25}$

Step 5.2

Evaluate the limit of

x

$x$ by plugging in

5

$5$ for

x

$x$ .

\frac{5^{2} - 3 \cdot 5 - 1 \cdot 10}{x^{2} - 10 x + 25}

$\frac{5^{2} - 3 \cdot 5 - 1 \cdot 10}{x^{2} - 10 x + 25}$

\frac{5^{2} - 3 \cdot 5 - 1 \cdot 10}{x^{2} - 10 x + 25}

$\frac{5^{2} - 3 \cdot 5 - 1 \cdot 10}{x^{2} - 10 x + 25}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise

5

$5$ to the power of

2

$2$ .

\frac{25 - 3 \cdot 5 - 1 \cdot 10}{x^{2} - 10 x + 25}

$\frac{25 - 3 \cdot 5 - 1 \cdot 10}{x^{2} - 10 x + 25}$

Step 6.1.2

Multiply

- 3

$- 3$ by

5

$5$ .

\frac{25 - 15 - 1 \cdot 10}{x^{2} - 10 x + 25}

$\frac{25 - 15 - 1 \cdot 10}{x^{2} - 10 x + 25}$

Step 6.1.3

Multiply

- 1

$- 1$ by

10

$10$ .

\frac{25 - 15 - 10}{x^{2} - 10 x + 25}

$\frac{25 - 15 - 10}{x^{2} - 10 x + 25}$

Step 6.1.4

Subtract

15

$15$ from

25

$25$ .

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

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

Step 6.1.5

Subtract

10

$10$ from

10

$10$ .

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

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

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

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

Step 6.2

Factor using the perfect square rule.

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Step 6.2.1

Rewrite

25

$25$ as

5^{2}

$5^{2}$ .

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

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

Step 6.2.2

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

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

Step 6.2.3

Rewrite the polynomial.

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

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

Step 6.2.4

Factor using the perfect square trinomial rule

a^{2} - 2 a b + b^{2} = {(a - b)}^{2}

$a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where

a = x

$a = x$ and

b = 5

$b = 5$ .

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

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

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

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

Step 6.3

Divide

0

$0$ by

{(x - 5)}^{2}

${(x - 5)}^{2}$ .

0

$0$

0

$0$