Calculus Examples

$lim_{x \to 1} \frac{x^{3} + 3 x^{2} - 11 x + 21}{2 x^{3} - 3 x^{2} + 17 x - 12}$

Step 1

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

$\frac{lim_{x \to 1} x^{3} + 3 x^{2} - 11 x + 21}{lim_{x \to 1} 2 x^{3} - 3 x^{2} + 17 x - 12}$

Step 2

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

$\frac{lim_{x \to 1} x^{3} + lim_{x \to 1} 3 x^{2} - lim_{x \to 1} 11 x + lim_{x \to 1} 21}{lim_{x \to 1} 2 x^{3} - 3 x^{2} + 17 x - 12}$

Step 3

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

$\frac{{(lim_{x \to 1} x)}^{3} + lim_{x \to 1} 3 x^{2} - lim_{x \to 1} 11 x + lim_{x \to 1} 21}{lim_{x \to 1} 2 x^{3} - 3 x^{2} + 17 x - 12}$

Step 4

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

$\frac{{(lim_{x \to 1} x)}^{3} + 3 lim_{x \to 1} x^{2} - lim_{x \to 1} 11 x + lim_{x \to 1} 21}{lim_{x \to 1} 2 x^{3} - 3 x^{2} + 17 x - 12}$

Step 5

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

$\frac{{(lim_{x \to 1} x)}^{3} + 3 {(lim_{x \to 1} x)}^{2} - lim_{x \to 1} 11 x + lim_{x \to 1} 21}{lim_{x \to 1} 2 x^{3} - 3 x^{2} + 17 x - 12}$

Step 6

Move the term $11$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to 1} x)}^{3} + 3 {(lim_{x \to 1} x)}^{2} - 11 lim_{x \to 1} x + lim_{x \to 1} 21}{lim_{x \to 1} 2 x^{3} - 3 x^{2} + 17 x - 12}$

Step 7

Evaluate the limit of $21$ which is constant as $x$ approaches $1$ .

$\frac{{(lim_{x \to 1} x)}^{3} + 3 {(lim_{x \to 1} x)}^{2} - 11 lim_{x \to 1} x + 21}{lim_{x \to 1} 2 x^{3} - 3 x^{2} + 17 x - 12}$

Step 8

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

$\frac{{(lim_{x \to 1} x)}^{3} + 3 {(lim_{x \to 1} x)}^{2} - 11 lim_{x \to 1} x + 21}{lim_{x \to 1} 2 x^{3} - lim_{x \to 1} 3 x^{2} + lim_{x \to 1} 17 x - lim_{x \to 1} 12}$

Step 9

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

$\frac{{(lim_{x \to 1} x)}^{3} + 3 {(lim_{x \to 1} x)}^{2} - 11 lim_{x \to 1} x + 21}{2 lim_{x \to 1} x^{3} - lim_{x \to 1} 3 x^{2} + lim_{x \to 1} 17 x - lim_{x \to 1} 12}$

Step 10

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

$\frac{{(lim_{x \to 1} x)}^{3} + 3 {(lim_{x \to 1} x)}^{2} - 11 lim_{x \to 1} x + 21}{2 {(lim_{x \to 1} x)}^{3} - lim_{x \to 1} 3 x^{2} + lim_{x \to 1} 17 x - lim_{x \to 1} 12}$

Step 11

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

$\frac{{(lim_{x \to 1} x)}^{3} + 3 {(lim_{x \to 1} x)}^{2} - 11 lim_{x \to 1} x + 21}{2 {(lim_{x \to 1} x)}^{3} - 3 lim_{x \to 1} x^{2} + lim_{x \to 1} 17 x - lim_{x \to 1} 12}$

Step 12

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

$\frac{{(lim_{x \to 1} x)}^{3} + 3 {(lim_{x \to 1} x)}^{2} - 11 lim_{x \to 1} x + 21}{2 {(lim_{x \to 1} x)}^{3} - 3 {(lim_{x \to 1} x)}^{2} + lim_{x \to 1} 17 x - lim_{x \to 1} 12}$

Step 13

Move the term $17$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to 1} x)}^{3} + 3 {(lim_{x \to 1} x)}^{2} - 11 lim_{x \to 1} x + 21}{2 {(lim_{x \to 1} x)}^{3} - 3 {(lim_{x \to 1} x)}^{2} + 17 lim_{x \to 1} x - lim_{x \to 1} 12}$

Step 14

Evaluate the limit of $12$ which is constant as $x$ approaches $1$ .

$\frac{{(lim_{x \to 1} x)}^{3} + 3 {(lim_{x \to 1} x)}^{2} - 11 lim_{x \to 1} x + 21}{2 {(lim_{x \to 1} x)}^{3} - 3 {(lim_{x \to 1} x)}^{2} + 17 lim_{x \to 1} x - 1 \cdot 12}$

Step 15

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

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Step 15.1