Calculus Examples

$lim_{x \to 0} \frac{2 x^{2} - 5 x - 3}{x^{2} + x - 12}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} 2 x^{2} - 5 x - 3}{lim_{x \to 0} x^{2} + x - 12}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} 2 x^{2} - lim_{x \to 0} 5 x - lim_{x \to 0} 3}{lim_{x \to 0} x^{2} + x - 12}$

Step 3

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

$\frac{2 lim_{x \to 0} x^{2} - lim_{x \to 0} 5 x - lim_{x \to 0} 3}{lim_{x \to 0} x^{2} + x - 12}$

Step 4

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

$\frac{2 {(lim_{x \to 0} x)}^{2} - lim_{x \to 0} 5 x - lim_{x \to 0} 3}{lim_{x \to 0} x^{2} + x - 12}$

Step 5

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

$\frac{2 {(lim_{x \to 0} x)}^{2} - 5 lim_{x \to 0} x - lim_{x \to 0} 3}{lim_{x \to 0} x^{2} + x - 12}$

Step 6

Evaluate the limit of $3$ which is constant as $x$ approaches $0$ .

$\frac{2 {(lim_{x \to 0} x)}^{2} - 5 lim_{x \to 0} x - 1 \cdot 3}{lim_{x \to 0} x^{2} + x - 12}$

Step 7

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{2 {(lim_{x \to 0} x)}^{2} - 5 lim_{x \to 0} x - 1 \cdot 3}{lim_{x \to 0} x^{2} + lim_{x \to 0} x - lim_{x \to 0} 12}$

Step 8

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

$\frac{2 {(lim_{x \to 0} x)}^{2} - 5 lim_{x \to 0} x - 1 \cdot 3}{{(lim_{x \to 0} x)}^{2} + lim_{x \to 0} x - lim_{x \to 0} 12}$

Step 9

Evaluate the limit of $12$ which is constant as $x$ approaches $0$ .

$\frac{2 {(lim_{x \to 0} x)}^{2} - 5 lim_{x \to 0} x - 1 \cdot 3}{{(lim_{x \to 0} x)}^{2} + lim_{x \to 0} x - 1 \cdot 12}$

Step 10

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 10.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{2 \cdot 0^{2} - 5 lim_{x \to 0} x - 1 \cdot 3}{{(lim_{x \to 0} x)}^{2} + lim_{x \to 0} x - 1 \cdot 12}$

Step 10.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{2 \cdot 0^{2} - 5 \cdot 0 - 1 \cdot 3}{{(lim_{x \to 0} x)}^{2} + lim_{x \to 0} x - 1 \cdot 12}$

Step 10.3

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{2 \cdot 0^{2} - 5 \cdot 0 - 1 \cdot 3}{0^{2} + lim_{x \to 0} x - 1 \cdot 12}$

Step 10.4

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{2 \cdot 0^{2} - 5 \cdot 0 - 1 \cdot 3}{0^{2} + 0 - 1 \cdot 12}$

Step 11

Simplify the answer.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Raising $0$ to any positive power yields $0$ .

$\frac{2 \cdot 0 - 5 \cdot 0 - 1 \cdot 3}{0^{2} + 0 - 1 \cdot 12}$

Step 11.1.2

Multiply $2$ by $0$ .

$\frac{0 - 5 \cdot 0 - 1 \cdot 3}{0^{2} + 0 - 1 \cdot 12}$

Step 11.1.3

Multiply $- 5$ by $0$ .

$\frac{0 + 0 - 1 \cdot 3}{0^{2} + 0 - 1 \cdot 12}$

Step 11.1.4

Multiply $- 1$ by $3$ .

$\frac{0 + 0 - 3}{0^{2} + 0 - 1 \cdot 12}$

Step 11.1.5

Add $0$ and $0$ .

$\frac{0 - 3}{0^{2} + 0 - 1 \cdot 12}$

Step 11.1.6

Subtract $3$ from $0$ .

$\frac{- 3}{0^{2} + 0 - 1 \cdot 12}$

Step 11.2

Simplify the denominator.

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Step 11.2.1

Raising $0$ to any positive power yields $0$ .

$\frac{- 3}{0 + 0 - 1 \cdot 12}$

Step 11.2.2

Multiply $- 1$ by $12$ .

$\frac{- 3}{0 + 0 - 12}$

Step 11.2.3

Add $0$ and $0$ .

$\frac{- 3}{0 - 12}$

Step 11.2.4

Subtract $12$ from $0$ .

$\frac{- 3}{- 12}$

Step 11.3

Cancel the common factor of $- 3$ and $- 12$ .

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Step 11.3.1

Factor $- 3$ out of $- 3$ .

$\frac{- 3 (1)}{- 12}$

Step 11.3.2

Cancel the common factors.

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Step 11.3.2.1

Factor $- 3$ out of $- 12$ .

$\frac{- 3 \cdot 1}{- 3 \cdot 4}$

Step 11.3.2.2

Cancel the common factor.

$\frac{- 3 \cdot 1}{- 3 \cdot 4}$

Step 11.3.2.3

Rewrite the expression.

$\frac{1}{4}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{4}$

Decimal Form:

$0.25$