Calculus Examples

$\frac{lim_{x \to 3} 2 x^{3} - 5 x^{2} - 2 x - 3}{4 x^{3} - 13 x^{2} + 4 x - 3}$

Step 1

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

$\frac{lim_{x \to 3} 2 x^{3} - lim_{x \to 3} 5 x^{2} - lim_{x \to 3} 2 x - lim_{x \to 3} 3}{4 x^{3} - 13 x^{2} + 4 x - 3}$

Step 2

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

$\frac{2 lim_{x \to 3} x^{3} - lim_{x \to 3} 5 x^{2} - lim_{x \to 3} 2 x - lim_{x \to 3} 3}{4 x^{3} - 13 x^{2} + 4 x - 3}$

Step 3

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

$\frac{2 {(lim_{x \to 3} x)}^{3} - lim_{x \to 3} 5 x^{2} - lim_{x \to 3} 2 x - lim_{x \to 3} 3}{4 x^{3} - 13 x^{2} + 4 x - 3}$

Step 4

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

$\frac{2 {(lim_{x \to 3} x)}^{3} - 5 lim_{x \to 3} x^{2} - lim_{x \to 3} 2 x - lim_{x \to 3} 3}{4 x^{3} - 13 x^{2} + 4 x - 3}$

Step 5

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

$\frac{2 {(lim_{x \to 3} x)}^{3} - 5 {(lim_{x \to 3} x)}^{2} - lim_{x \to 3} 2 x - lim_{x \to 3} 3}{4 x^{3} - 13 x^{2} + 4 x - 3}$

Step 6

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

$\frac{2 {(lim_{x \to 3} x)}^{3} - 5 {(lim_{x \to 3} x)}^{2} - 2 lim_{x \to 3} x - lim_{x \to 3} 3}{4 x^{3} - 13 x^{2} + 4 x - 3}$

Step 7

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

$\frac{2 {(lim_{x \to 3} x)}^{3} - 5 {(lim_{x \to 3} x)}^{2} - 2 lim_{x \to 3} x - 1 \cdot 3}{4 x^{3} - 13 x^{2} + 4 x - 3}$

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 $x$ .

$\frac{2 \cdot 3^{3} - 5 {(lim_{x \to 3} x)}^{2} - 2 lim_{x \to 3} x - 1 \cdot 3}{4 x^{3} - 13 x^{2} + 4 x - 3}$

Step 8.2

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

$\frac{2 \cdot 3^{3} - 5 \cdot 3^{2} - 2 lim_{x \to 3} x - 1 \cdot 3}{4 x^{3} - 13 x^{2} + 4 x - 3}$

Step 8.3

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

$\frac{2 \cdot 3^{3} - 5 \cdot 3^{2} - 2 \cdot 3 - 1 \cdot 3}{4 x^{3} - 13 x^{2} + 4 x - 3}$

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 $3$ .

$\frac{2 \cdot 27 - 5 \cdot 3^{2} - 2 \cdot 3 - 1 \cdot 3}{4 x^{3} - 13 x^{2} + 4 x - 3}$

Step 9.1.2

Multiply $2$ by $27$ .

$\frac{54 - 5 \cdot 3^{2} - 2 \cdot 3 - 1 \cdot 3}{4 x^{3} - 13 x^{2} + 4 x - 3}$

Step 9.1.3

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

$\frac{54 - 5 \cdot 9 - 2 \cdot 3 - 1 \cdot 3}{4 x^{3} - 13 x^{2} + 4 x - 3}$

Step 9.1.4

Multiply $- 5$ by $9$ .

$\frac{54 - 45 - 2 \cdot 3 - 1 \cdot 3}{4 x^{3} - 13 x^{2} + 4 x - 3}$

Step 9.1.5

Multiply $- 2$ by $3$ .

$\frac{54 - 45 - 6 - 1 \cdot 3}{4 x^{3} - 13 x^{2} + 4 x - 3}$

Step 9.1.6

Multiply $- 1$ by $3$ .

$\frac{54 - 45 - 6 - 3}{4 x^{3} - 13 x^{2} + 4 x - 3}$

Step 9.1.7

Subtract $45$ from $54$ .

$\frac{9 - 6 - 3}{4 x^{3} - 13 x^{2} + 4 x - 3}$

Step 9.1.8

Subtract $6$ from $9$ .

$\frac{3 - 3}{4 x^{3} - 13 x^{2} + 4 x - 3}$

Step 9.1.9

Subtract $3$ from $3$ .

$\frac{0}{4 x^{3} - 13 x^{2} + 4 x - 3}$

Step 9.2

Divide $0$ by $4 x^{3} - 13 x^{2} + 4 x - 3$ .

$0$