Calculus Examples

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

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

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

Step 2

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

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

Step 3

Evaluate the limit of $6$ which is constant as $x$ approaches $2$ .

$\frac{{(lim_{x \to 2} x)}^{2} + lim_{x \to 2} x - 1 \cdot 6}{x - 2}$

Step 4

Evaluate the limits by plugging in $2$ for all occurrences of $x$ .

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Step 4.1

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{2^{2} + lim_{x \to 2} x - 1 \cdot 6}{x - 2}$

Step 4.2

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{2^{2} + 2 - 1 \cdot 6}{x - 2}$

Step 5

Simplify the answer.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

$\frac{4 + 2 - 1 \cdot 6}{x - 2}$

Step 5.1.2

Multiply $- 1$ by $6$ .

$\frac{4 + 2 - 6}{x - 2}$

Step 5.1.3

Add $4$ and $2$ .

$\frac{6 - 6}{x - 2}$

Step 5.1.4

Subtract $6$ from $6$ .

$\frac{0}{x - 2}$

Step 5.2

Divide $0$ by $x - 2$ .

$0$