Calculus Examples

$\frac{lim_{x \to 8} 5 x^{3} - 3 x^{2} + 10}{x^{2}}$

Step 1

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

$\frac{lim_{x \to 8} 5 x^{3} - lim_{x \to 8} 3 x^{2} + lim_{x \to 8} 10}{x^{2}}$

Step 2

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

$\frac{5 lim_{x \to 8} x^{3} - lim_{x \to 8} 3 x^{2} + lim_{x \to 8} 10}{x^{2}}$

Step 3

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

$\frac{5 {(lim_{x \to 8} x)}^{3} - lim_{x \to 8} 3 x^{2} + lim_{x \to 8} 10}{x^{2}}$

Step 4

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

$\frac{5 {(lim_{x \to 8} x)}^{3} - 3 lim_{x \to 8} x^{2} + lim_{x \to 8} 10}{x^{2}}$

Step 5

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

$\frac{5 {(lim_{x \to 8} x)}^{3} - 3 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} 10}{x^{2}}$

Step 6

Evaluate the limit of $10$ which is constant as $x$ approaches $8$ .

$\frac{5 {(lim_{x \to 8} x)}^{3} - 3 {(lim_{x \to 8} x)}^{2} + 10}{x^{2}}$

Step 7

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

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Step 7.1

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

$\frac{5 \cdot 8^{3} - 3 {(lim_{x \to 8} x)}^{2} + 10}{x^{2}}$

Step 7.2

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

$\frac{5 \cdot 8^{3} - 3 \cdot 8^{2} + 10}{x^{2}}$

Step 8

Simplify the numerator.

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Step 8.1

Raise $8$ to the power of $3$ .

$\frac{5 \cdot 512 - 3 \cdot 8^{2} + 10}{x^{2}}$

Step 8.2

Multiply $5$ by $512$ .

$\frac{2560 - 3 \cdot 8^{2} + 10}{x^{2}}$

Step 8.3

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

$\frac{2560 - 3 \cdot 64 + 10}{x^{2}}$

Step 8.4

Multiply $- 3$ by $64$ .

$\frac{2560 - 192 + 10}{x^{2}}$

Step 8.5

Subtract $192$ from $2560$ .

$\frac{2368 + 10}{x^{2}}$

Step 8.6

Add $2368$ and $10$ .

$\frac{2378}{x^{2}}$