Calculus Examples

$\frac{lim_{x \to 2} x^{2} - 4 x + 4}{x^{3} + 5 x^{2} - 14 x}$

Step 1

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

$\frac{lim_{x \to 2} x^{2} - lim_{x \to 2} 4 x + lim_{x \to 2} 4}{x^{3} + 5 x^{2} - 14 x}$

Step 2

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

$\frac{{(lim_{x \to 2} x)}^{2} - lim_{x \to 2} 4 x + lim_{x \to 2} 4}{x^{3} + 5 x^{2} - 14 x}$

Step 3

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to 2} x)}^{2} - 4 lim_{x \to 2} x + lim_{x \to 2} 4}{x^{3} + 5 x^{2} - 14 x}$

Step 4

Evaluate the limit of $4$ which is constant as $x$ approaches $2$ .

$\frac{{(lim_{x \to 2} x)}^{2} - 4 lim_{x \to 2} x + 4}{x^{3} + 5 x^{2} - 14 x}$

Step 5

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

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Step 5.1

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

$\frac{2^{2} - 4 lim_{x \to 2} x + 4}{x^{3} + 5 x^{2} - 14 x}$

Step 5.2

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

$\frac{2^{2} - 4 \cdot 2 + 4}{x^{3} + 5 x^{2} - 14 x}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

$\frac{4 - 4 \cdot 2 + 4}{x^{3} + 5 x^{2} - 14 x}$

Step 6.1.2

Multiply $- 4$ by $2$ .

$\frac{4 - 8 + 4}{x^{3} + 5 x^{2} - 14 x}$

Step 6.1.3

Subtract $8$ from $4$ .

$\frac{- 4 + 4}{x^{3} + 5 x^{2} - 14 x}$

Step 6.1.4

Add $- 4$ and $4$ .

$\frac{0}{x^{3} + 5 x^{2} - 14 x}$

Step 6.2

Simplify the denominator.

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Step 6.2.1

Factor $x$ out of $x^{3} + 5 x^{2} - 14 x$ .

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Step 6.2.1.1

Factor $x$ out of $x^{3}$ .

$\frac{0}{x \cdot x^{2} + 5 x^{2} - 14 x}$

Step 6.2.1.2

Factor $x$ out of $5 x^{2}$ .

$\frac{0}{x \cdot x^{2} + x (5 x) - 14 x}$

Step 6.2.1.3

Factor $x$ out of $- 14 x$ .

$\frac{0}{x \cdot x^{2} + x (5 x) + x \cdot - 14}$

Step 6.2.1.4

Factor $x$ out of $x \cdot x^{2} + x (5 x)$ .

$\frac{0}{x (x^{2} + 5 x) + x \cdot - 14}$

Step 6.2.1.5

Factor $x$ out of $x (x^{2} + 5 x) + x \cdot - 14$ .

$\frac{0}{x (x^{2} + 5 x - 14)}$

Step 6.2.2

Factor $x^{2} + 5 x - 14$ using the AC method.

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Step 6.2.2.1

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 $- 14$ and whose sum is $5$ .

$- 2, 7$

Step 6.2.2.2

Write the factored form using these integers.

$\frac{0}{x ((x - 2) (x + 7))}$

$\frac{0}{x (x - 2) (x + 7)}$

Step 6.3

Divide $0$ by $x (x - 2) (x + 7)$ .

$0$