Calculus Examples

$lim_{x \to 8} \frac{3 x - 2 x^{2} + 4 x^{3}}{7 - 2 x - x^{3}}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $8$ .

$\frac{lim_{x \to 8} 3 x - 2 x^{2} + 4 x^{3}}{lim_{x \to 8} 7 - 2 x - x^{3}}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{lim_{x \to 8} 3 x - lim_{x \to 8} 2 x^{2} + lim_{x \to 8} 4 x^{3}}{lim_{x \to 8} 7 - 2 x - x^{3}}$

Step 3

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

$\frac{3 lim_{x \to 8} x - lim_{x \to 8} 2 x^{2} + lim_{x \to 8} 4 x^{3}}{lim_{x \to 8} 7 - 2 x - x^{3}}$

Step 4

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

$\frac{3 lim_{x \to 8} x - 2 lim_{x \to 8} x^{2} + lim_{x \to 8} 4 x^{3}}{lim_{x \to 8} 7 - 2 x - x^{3}}$

Step 5

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

$\frac{3 lim_{x \to 8} x - 2 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} 4 x^{3}}{lim_{x \to 8} 7 - 2 x - x^{3}}$

Step 6

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

$\frac{3 lim_{x \to 8} x - 2 {(lim_{x \to 8} x)}^{2} + 4 lim_{x \to 8} x^{3}}{lim_{x \to 8} 7 - 2 x - x^{3}}$

Step 7

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

$\frac{3 lim_{x \to 8} x - 2 {(lim_{x \to 8} x)}^{2} + 4 {(lim_{x \to 8} x)}^{3}}{lim_{x \to 8} 7 - 2 x - x^{3}}$

Step 8

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{3 lim_{x \to 8} x - 2 {(lim_{x \to 8} x)}^{2} + 4 {(lim_{x \to 8} x)}^{3}}{lim_{x \to 8} 7 - lim_{x \to 8} 2 x - lim_{x \to 8} x^{3}}$

Step 9

Evaluate the limit of $7$ which is constant as $x$ approaches $8$ .

$\frac{3 lim_{x \to 8} x - 2 {(lim_{x \to 8} x)}^{2} + 4 {(lim_{x \to 8} x)}^{3}}{7 - lim_{x \to 8} 2 x - lim_{x \to 8} x^{3}}$

Step 10

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

$\frac{3 lim_{x \to 8} x - 2 {(lim_{x \to 8} x)}^{2} + 4 {(lim_{x \to 8} x)}^{3}}{7 - 2 lim_{x \to 8} x - lim_{x \to 8} x^{3}}$

Step 11

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

$\frac{3 lim_{x \to 8} x - 2 {(lim_{x \to 8} x)}^{2} + 4 {(lim_{x \to 8} x)}^{3}}{7 - 2 lim_{x \to 8} x - {(lim_{x \to 8} x)}^{3}}$

Step 12

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

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Step 12.1