Calculus Examples

$lim_{x \to 2} \frac{x^{3} - 7 x}{x^{2} + 3 x}$

Step 1

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

$\frac{lim_{x \to 2} x^{3} - 7 x}{lim_{x \to 2} x^{2} + 3 x}$

Step 2

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

$\frac{lim_{x \to 2} x^{3} - lim_{x \to 2} 7 x}{lim_{x \to 2} x^{2} + 3 x}$

Step 3

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 2} x)}^{3} - lim_{x \to 2} 7 x}{lim_{x \to 2} x^{2} + 3 x}$

Step 4

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to 2} x)}^{3} - 7 lim_{x \to 2} x}{lim_{x \to 2} x^{2} + 3 x}$

Step 5

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

$\frac{{(lim_{x \to 2} x)}^{3} - 7 lim_{x \to 2} x}{lim_{x \to 2} x^{2} + lim_{x \to 2} 3 x}$

Step 6

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

$\frac{{(lim_{x \to 2} x)}^{3} - 7 lim_{x \to 2} x}{{(lim_{x \to 2} x)}^{2} + lim_{x \to 2} 3 x}$

Step 7

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to 2} x)}^{3} - 7 lim_{x \to 2} x}{{(lim_{x \to 2} x)}^{2} + 3 lim_{x \to 2} x}$

Step 8

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

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Step 8.1

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

$\frac{2^{3} - 7 lim_{x \to 2} x}{{(lim_{x \to 2} x)}^{2} + 3 lim_{x \to 2} x}$

Step 8.2

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

$\frac{2^{3} - 7 \cdot 2}{{(lim_{x \to 2} x)}^{2} + 3 lim_{x \to 2} x}$

Step 8.3

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

$\frac{2^{3} - 7 \cdot 2}{2^{2} + 3 lim_{x \to 2} x}$

Step 8.4

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

$\frac{2^{3} - 7 \cdot 2}{2^{2} + 3 \cdot 2}$

Step 9

Simplify the answer.

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Step 9.1

Cancel the common factor of $2^{3} - 7 \cdot 2$ and $2^{2} + 3 \cdot 2$ .

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Step 9.1.1

Factor $2$ out of $2^{3}$ .

$\frac{2 \cdot 2^{2} - 7 \cdot 2}{2^{2} + 3 \cdot 2}$

Step 9.1.2

Factor $2$ out of $- 7 \cdot 2$ .

$\frac{2 \cdot 2^{2} + 2 \cdot - 7}{2^{2} + 3 \cdot 2}$

Step 9.1.3

Factor $2$ out of $2 \cdot 2^{2} + 2 \cdot - 7$ .

$\frac{2 \cdot (2^{2} - 7)}{2^{2} + 3 \cdot 2}$

Step 9.1.4

Cancel the common factors.

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Step 9.1.4.1

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

$\frac{2 \cdot (2^{2} - 7)}{2 \cdot 2 + 3 \cdot 2}$

Step 9.1.4.2

Factor $2$ out of $3 \cdot 2$ .

$\frac{2 \cdot (2^{2} - 7)}{2 \cdot 2 + 2 \cdot 3}$

Step 9.1.4.3

Factor $2$ out of $2 \cdot 2 + 2 \cdot 3$ .

$\frac{2 \cdot (2^{2} - 7)}{2 \cdot (2 + 3)}$

Step 9.1.4.4

Cancel the common factor.

$\frac{2 \cdot (2^{2} - 7)}{2 \cdot (2 + 3)}$

Step 9.1.4.5

Rewrite the expression.

$\frac{2^{2} - 7}{2 + 3}$

Step 9.2

Simplify the numerator.

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Step 9.2.1

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

$\frac{4 - 7}{2 + 3}$

Step 9.2.2

Subtract $7$ from $4$ .

$\frac{- 3}{2 + 3}$

Step 9.3

Add $2$ and $3$ .

$\frac{- 3}{5}$

Step 9.4

Move the negative in front of the fraction.

$- \frac{3}{5}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- \frac{3}{5}$

Decimal Form:

$- 0.6$