Calculus Examples

$\frac{lim_{x \to 5} x^{2} - 6 x + 5}{x - 5}$

Step 1

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

$\frac{lim_{x \to 5} x^{2} - lim_{x \to 5} 6 x + lim_{x \to 5} 5}{x - 5}$

Step 2

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

$\frac{{(lim_{x \to 5} x)}^{2} - lim_{x \to 5} 6 x + lim_{x \to 5} 5}{x - 5}$

Step 3

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to 5} x)}^{2} - 6 lim_{x \to 5} x + lim_{x \to 5} 5}{x - 5}$

Step 4

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

$\frac{{(lim_{x \to 5} x)}^{2} - 6 lim_{x \to 5} x + 5}{x - 5}$

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 $x$ .

$\frac{5^{2} - 6 lim_{x \to 5} x + 5}{x - 5}$

Step 5.2

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

$\frac{5^{2} - 6 \cdot 5 + 5}{x - 5}$

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 $2$ .

$\frac{25 - 6 \cdot 5 + 5}{x - 5}$

Step 6.1.2

Multiply $- 6$ by $5$ .

$\frac{25 - 30 + 5}{x - 5}$

Step 6.1.3

Subtract $30$ from $25$ .

$\frac{- 5 + 5}{x - 5}$

Step 6.1.4

Add $- 5$ and $5$ .

$\frac{0}{x - 5}$

Step 6.2

Divide $0$ by $x - 5$ .

$0$