Calculus Examples

$lim_{x \to - 8} (- 27 x^{38} + 2 x^{2} - 5 x) - \frac{5}{3}$

Step 1

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

$- lim_{x \to - 8} 27 x^{38} + lim_{x \to - 8} 2 x^{2} - lim_{x \to - 8} 5 x - lim_{x \to - 8} \frac{5}{3}$

Step 2

Move the term $27$ outside of the limit because it is constant with respect to $x$ .

$- 27 lim_{x \to - 8} x^{38} + lim_{x \to - 8} 2 x^{2} - lim_{x \to - 8} 5 x - lim_{x \to - 8} \frac{5}{3}$

Step 3

Move the exponent $38$ from $x^{38}$ outside the limit using the Limits Power Rule.

$- 27 {(lim_{x \to - 8} x)}^{38} + lim_{x \to - 8} 2 x^{2} - lim_{x \to - 8} 5 x - lim_{x \to - 8} \frac{5}{3}$

Step 4

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

$- 27 {(lim_{x \to - 8} x)}^{38} + 2 lim_{x \to - 8} x^{2} - lim_{x \to - 8} 5 x - lim_{x \to - 8} \frac{5}{3}$

Step 5

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

$- 27 {(lim_{x \to - 8} x)}^{38} + 2 {(lim_{x \to - 8} x)}^{2} - lim_{x \to - 8} 5 x - lim_{x \to - 8} \frac{5}{3}$

Step 6

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

$- 27 {(lim_{x \to - 8} x)}^{38} + 2 {(lim_{x \to - 8} x)}^{2} - 5 lim_{x \to - 8} x - lim_{x \to - 8} \frac{5}{3}$

Step 7

Evaluate the limit of $\frac{5}{3}$ which is constant as $x$ approaches $- 8$ .

$- 27 {(lim_{x \to - 8} x)}^{38} + 2 {(lim_{x \to - 8} x)}^{2} - 5 lim_{x \to - 8} x - \frac{5}{3}$

Step 8

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

Tap for more steps...

Step 8.1

Evaluate the limit of $x$ by plugging in $- 8$ for $x$ .

$- 27 {(- 8)}^{38} + 2 {(lim_{x \to - 8} x)}^{2} - 5 lim_{x \to - 8} x - \frac{5}{3}$

Step 8.2

Evaluate the limit of $x$ by plugging in $- 8$ for $x$ .

$- 27 {(- 8)}^{38} + 2 {(- 8)}^{2} - 5 lim_{x \to - 8} x - \frac{5}{3}$

Step 8.3

Evaluate the limit of $x$ by plugging in $- 8$ for $x$ .

$- 27 {(- 8)}^{38} + 2 {(- 8)}^{2} - 5 \cdot - 8 - \frac{5}{3}$

Step 9

Simplify the answer.

Tap for more steps...

Step 9.1

Simplify each term.

Tap for more steps...

Step 9.1.1

Raise $- 8$ to the power of $2$ .

$- 27 {(- 8)}^{38} + 2 \cdot 64 - 5 \cdot - 8 - \frac{5}{3}$

Step 9.1.2

Multiply $2$ by $64$ .

$- 27 {(- 8)}^{38} + 128 - 5 \cdot - 8 - \frac{5}{3}$

Step 9.1.3

Multiply $- 5$ by $- 8$ .

$- 27 {(- 8)}^{38} + 128 + 40 - \frac{5}{3}$

Step 9.2

Add $128$ and $40$ .

$- 27 {(- 8)}^{38} + 168 - \frac{5}{3}$

Step 9.3

To write $168$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- 27 {(- 8)}^{38} + 168 \cdot \frac{3}{3} - \frac{5}{3}$

Step 9.4

Combine $168$ and $\frac{3}{3}$ .

$- 27 {(- 8)}^{38} + \frac{168 \cdot 3}{3} - \frac{5}{3}$

Step 9.5

Combine the numerators over the common denominator.

$- 27 {(- 8)}^{38} + \frac{168 \cdot 3 - 5}{3}$

Step 9.6

Simplify the numerator.

Tap for more steps...

Step 9.6.1

Multiply $168$ by $3$ .

$- 27 {(- 8)}^{38} + \frac{504 - 5}{3}$

Step 9.6.2

Subtract $5$ from $504$ .

$- 27 {(- 8)}^{38} + \frac{499}{3}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- 27 {(- 8)}^{38} + \frac{499}{3}$

Decimal Form:

$- 5.6076806 \cdot 10^{35}$