Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

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

Step 4

Evaluate the limit of $10$ which is constant as $x$ approaches $5$ .

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

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

Step 6.1.2

Multiply $- 3$ by $5$ .

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

Step 6.1.3

Multiply $- 1$ by $10$ .

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

Step 6.1.4

Subtract $15$ from $25$ .

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

Step 6.1.5

Subtract $10$ from $10$ .

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

Step 6.2

Factor using the perfect square rule.

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Step 6.2.1

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

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

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.

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

Step 6.2.3

Rewrite the polynomial.

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

Step 6.2.4

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

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

Step 6.3

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

$0$