Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Evaluate the limit of $5$ which is constant as $x$ approaches $8$ .

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

$\frac{8^{2} + 2 \cdot 8 - 1 \cdot 5}{3 x^{2} + 2}$

Step 6

Simplify the numerator.

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Step 6.1

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

$\frac{64 + 2 \cdot 8 - 1 \cdot 5}{3 x^{2} + 2}$

Step 6.2

Multiply $2$ by $8$ .

$\frac{64 + 16 - 1 \cdot 5}{3 x^{2} + 2}$

Step 6.3

Multiply $- 1$ by $5$ .

$\frac{64 + 16 - 5}{3 x^{2} + 2}$

Step 6.4

Add $64$ and $16$ .

$\frac{80 - 5}{3 x^{2} + 2}$

Step 6.5

Subtract $5$ from $80$ .

$\frac{75}{3 x^{2} + 2}$