Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Evaluate the limit of $6$ which is constant as $x$ approaches $5$ .

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

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 $x$ .

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

Step 5.2

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

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

Step 6

Simplify the numerator.

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Step 6.1

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

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

Step 6.2

Multiply $- 5$ by $5$ .

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

Step 6.3

Subtract $25$ from $25$ .

$\frac{0 + 6}{x - 5}$

Step 6.4

Add $0$ and $6$ .

$\frac{6}{x - 5}$