Calculus Examples

$lim_{x \to 2} \frac{x^{2} - 4 x + 11}{x^{3} - 8 x^{2} + 19}$

Step 1

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

$\frac{lim_{x \to 2} x^{2} - 4 x + 11}{lim_{x \to 2} x^{3} - 8 x^{2} + 19}$

Step 2

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

$\frac{lim_{x \to 2} x^{2} - lim_{x \to 2} 4 x + lim_{x \to 2} 11}{lim_{x \to 2} x^{3} - 8 x^{2} + 19}$

Step 3

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

$\frac{{(lim_{x \to 2} x)}^{2} - lim_{x \to 2} 4 x + lim_{x \to 2} 11}{lim_{x \to 2} x^{3} - 8 x^{2} + 19}$

Step 4

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

$\frac{{(lim_{x \to 2} x)}^{2} - 4 lim_{x \to 2} x + lim_{x \to 2} 11}{lim_{x \to 2} x^{3} - 8 x^{2} + 19}$

Step 5

Evaluate the limit of $11$ which is constant as $x$ approaches $2$ .

$\frac{{(lim_{x \to 2} x)}^{2} - 4 lim_{x \to 2} x + 11}{lim_{x \to 2} x^{3} - 8 x^{2} + 19}$

Step 6

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

$\frac{{(lim_{x \to 2} x)}^{2} - 4 lim_{x \to 2} x + 11}{lim_{x \to 2} x^{3} - lim_{x \to 2} 8 x^{2} + lim_{x \to 2} 19}$

Step 7

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

$\frac{{(lim_{x \to 2} x)}^{2} - 4 lim_{x \to 2} x + 11}{{(lim_{x \to 2} x)}^{3} - lim_{x \to 2} 8 x^{2} + lim_{x \to 2} 19}$

Step 8

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to 2} x)}^{2} - 4 lim_{x \to 2} x + 11}{{(lim_{x \to 2} x)}^{3} - 8 lim_{x \to 2} x^{2} + lim_{x \to 2} 19}$

Step 9

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

$\frac{{(lim_{x \to 2} x)}^{2} - 4 lim_{x \to 2} x + 11}{{(lim_{x \to 2} x)}^{3} - 8 {(lim_{x \to 2} x)}^{2} + lim_{x \to 2} 19}$

Step 10

Evaluate the limit of $19$ which is constant as $x$ approaches $2$ .

$\frac{{(lim_{x \to 2} x)}^{2} - 4 lim_{x \to 2} x + 11}{{(lim_{x \to 2} x)}^{3} - 8 {(lim_{x \to 2} x)}^{2} + 19}$

Step 11

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

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Step 11.1

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

$\frac{2^{2} - 4 lim_{x \to 2} x + 11}{{(lim_{x \to 2} x)}^{3} - 8 {(lim_{x \to 2} x)}^{2} + 19}$

Step 11.2

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

$\frac{2^{2} - 4 \cdot 2 + 11}{{(lim_{x \to 2} x)}^{3} - 8 {(lim_{x \to 2} x)}^{2} + 19}$

Step 11.3

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

$\frac{2^{2} - 4 \cdot 2 + 11}{2^{3} - 8 {(lim_{x \to 2} x)}^{2} + 19}$

Step 11.4

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

$\frac{2^{2} - 4 \cdot 2 + 11}{2^{3} - 8 \cdot 2^{2} + 19}$

Step 12

Simplify the answer.

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Step 12.1

Simplify the numerator.

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Step 12.1.1

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

$\frac{4 - 4 \cdot 2 + 11}{2^{3} - 8 \cdot 2^{2} + 19}$

Step 12.1.2

Multiply $- 4$ by $2$ .

$\frac{4 - 8 + 11}{2^{3} - 8 \cdot 2^{2} + 19}$

Step 12.1.3

Subtract $8$ from $4$ .

$\frac{- 4 + 11}{2^{3} - 8 \cdot 2^{2} + 19}$

Step 12.1.4

Add $- 4$ and $11$ .

$\frac{7}{2^{3} - 8 \cdot 2^{2} + 19}$

Step 12.2

Simplify the denominator.

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Step 12.2.1

Raise $2$ to the power of $3$ .

$\frac{7}{8 - 8 \cdot 2^{2} + 19}$

Step 12.2.2

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

$\frac{7}{8 - 8 \cdot 4 + 19}$

Step 12.2.3

Multiply $- 8$ by $4$ .

$\frac{7}{8 - 32 + 19}$

Step 12.2.4

Subtract $32$ from $8$ .

$\frac{7}{- 24 + 19}$

Step 12.2.5

Add $- 24$ and $19$ .

$\frac{7}{- 5}$

Step 12.3

Move the negative in front of the fraction.

$- \frac{7}{5}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$- \frac{7}{5}$

Decimal Form:

$- 1.4$