Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Move the term $2$ outside of the limit because it is constant with respect to $u$ .

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

Step 5

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

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

Step 6

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

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Step 6.1

Evaluate the limit of $u$ by plugging in $- 2$ for $u$ .

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

Step 6.2

Evaluate the limit of $u$ by plugging in $- 2$ for $u$ .

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

Step 6.3

Evaluate the limit of $u$ by plugging in $- 2$ for $u$ .

$\frac{{(- 2)}^{2} - x \cdot - 2 + 2 \cdot - 2 - 2 x}{u^{2} - u - 6}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

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

Step 7.1.2

Multiply $- 2$ by $- 1$ .

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

Step 7.1.3

Multiply $2$ by $- 2$ .

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

Step 7.1.4

Subtract $4$ from $4$ .

$\frac{2 x + 0 - 2 x}{u^{2} - u - 6}$

Step 7.1.5

Add $2 x$ and $0$ .

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

Step 7.1.6

Subtract $2 x$ from $2 x$ .

$\frac{0}{u^{2} - u - 6}$

Step 7.2

Factor $u^{2} - u - 6$ using the AC method.

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Step 7.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Step 7.2.2

Write the factored form using these integers.

$\frac{0}{(u - 3) (u + 2)}$

Step 7.3

Divide $0$ by $(u - 3) (u + 2)$ .

$0$