Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

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

Step 4

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

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

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 $x$ .

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

Step 5.2

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

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

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 $2$ .

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

Step 6.1.2

Multiply $- 5$ by $5$ .

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

Step 6.1.3

Subtract $25$ from $25$ .

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

Step 6.1.4

Add $0$ and $10$ .

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

Step 6.2

Simplify the denominator.

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Step 6.2.1

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

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

Step 6.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 5$ .

$\frac{10}{(x + 5) (x - 5)}$