Calculus Examples

$lim_{x \to 8} \frac{1 + x + x^{3} - 5 x^{5}}{3 x^{5} + 1000 x^{3} - 2}$

Step 1

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

$\frac{lim_{x \to 8} 1 + x + x^{3} - 5 x^{5}}{lim_{x \to 8} 3 x^{5} + 1000 x^{3} - 2}$

Step 2

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

$\frac{lim_{x \to 8} 1 + lim_{x \to 8} x + lim_{x \to 8} x^{3} - lim_{x \to 8} 5 x^{5}}{lim_{x \to 8} 3 x^{5} + 1000 x^{3} - 2}$

Step 3

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

$\frac{1 + lim_{x \to 8} x + lim_{x \to 8} x^{3} - lim_{x \to 8} 5 x^{5}}{lim_{x \to 8} 3 x^{5} + 1000 x^{3} - 2}$

Step 4

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

$\frac{1 + lim_{x \to 8} x + {(lim_{x \to 8} x)}^{3} - lim_{x \to 8} 5 x^{5}}{lim_{x \to 8} 3 x^{5} + 1000 x^{3} - 2}$

Step 5

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

$\frac{1 + lim_{x \to 8} x + {(lim_{x \to 8} x)}^{3} - 5 lim_{x \to 8} x^{5}}{lim_{x \to 8} 3 x^{5} + 1000 x^{3} - 2}$

Step 6

Move the exponent $5$ from $x^{5}$ outside the limit using the Limits Power Rule.

$\frac{1 + lim_{x \to 8} x + {(lim_{x \to 8} x)}^{3} - 5 {(lim_{x \to 8} x)}^{5}}{lim_{x \to 8} 3 x^{5} + 1000 x^{3} - 2}$

Step 7

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

$\frac{1 + lim_{x \to 8} x + {(lim_{x \to 8} x)}^{3} - 5 {(lim_{x \to 8} x)}^{5}}{lim_{x \to 8} 3 x^{5} + lim_{x \to 8} 1000 x^{3} - lim_{x \to 8} 2}$

Step 8

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

$\frac{1 + lim_{x \to 8} x + {(lim_{x \to 8} x)}^{3} - 5 {(lim_{x \to 8} x)}^{5}}{3 lim_{x \to 8} x^{5} + lim_{x \to 8} 1000 x^{3} - lim_{x \to 8} 2}$

Step 9

Move the exponent $5$ from $x^{5}$ outside the limit using the Limits Power Rule.

$\frac{1 + lim_{x \to 8} x + {(lim_{x \to 8} x)}^{3} - 5 {(lim_{x \to 8} x)}^{5}}{3 {(lim_{x \to 8} x)}^{5} + lim_{x \to 8} 1000 x^{3} - lim_{x \to 8} 2}$

Step 10

Move the term $1000$ outside of the limit because it is constant with respect to $x$ .

$\frac{1 + lim_{x \to 8} x + {(lim_{x \to 8} x)}^{3} - 5 {(lim_{x \to 8} x)}^{5}}{3 {(lim_{x \to 8} x)}^{5} + 1000 lim_{x \to 8} x^{3} - lim_{x \to 8} 2}$

Step 11

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

$\frac{1 + lim_{x \to 8} x + {(lim_{x \to 8} x)}^{3} - 5 {(lim_{x \to 8} x)}^{5}}{3 {(lim_{x \to 8} x)}^{5} + 1000 {(lim_{x \to 8} x)}^{3} - lim_{x \to 8} 2}$

Step 12

Evaluate the limit of $2$ which is constant as $x$ approaches $8$ .

$\frac{1 + lim_{x \to 8} x + {(lim_{x \to 8} x)}^{3} - 5 {(lim_{x \to 8} x)}^{5}}{3 {(lim_{x \to 8} x)}^{5} + 1000 {(lim_{x \to 8} x)}^{3} - 1 \cdot 2}$

Step 13

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

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Step 13.1