Calculus Examples

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

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $3$ .

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

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

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

Step 3

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

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

Step 4

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

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

Step 5

Evaluate the limit of $3$ which is constant as $x$ approaches $3$ .

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

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

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

Step 7

Evaluate the limit of $4$ which is constant as $x$ approaches $3$ .

$\frac{8 {(lim_{x \to 3} x)}^{2} + lim_{x \to 3} x + 3}{lim_{x \to 3} x - 1 \cdot 4}$

Step 8

Evaluate the limits by plugging in $3$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{8 \cdot 3^{2} + lim_{x \to 3} x + 3}{lim_{x \to 3} x - 1 \cdot 4}$

Step 8.2

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{8 \cdot 3^{2} + 3 + 3}{lim_{x \to 3} x - 1 \cdot 4}$

Step 8.3

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{8 \cdot 3^{2} + 3 + 3}{3 - 1 \cdot 4}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Raise $3$ to the power of $2$ .

$\frac{8 \cdot 9 + 3 + 3}{3 - 1 \cdot 4}$

Step 9.1.2

Multiply $8$ by $9$ .

$\frac{72 + 3 + 3}{3 - 1 \cdot 4}$

Step 9.1.3

Add $72$ and $3$ .

$\frac{75 + 3}{3 - 1 \cdot 4}$

Step 9.1.4

Add $75$ and $3$ .

$\frac{78}{3 - 1 \cdot 4}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

Multiply $- 1$ by $4$ .

$\frac{78}{3 - 4}$

Step 9.2.2

Subtract $4$ from $3$ .

$\frac{78}{- 1}$

Step 9.3

Divide $78$ by $- 1$ .

$- 78$