Calculus Examples

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

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $1$ .

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

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

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

Step 6

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

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

Step 7

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

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

Step 8

Evaluate the limit of $2$ which is constant as $x$ approaches $1$ .

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

Step 9

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

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

Step 10

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

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

Step 11

Evaluate the limit of $2$ which is constant as $x$ approaches $1$ .

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

Step 12

Evaluate the limits by plugging in $1$ for all occurrences of $x$ .

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Step 12.1