Calculus Examples

$lim_{x \to 1} \frac{x^{3} - 2 x^{2} - 2 x - 3}{x^{4} - 2 x^{3} - 27}$

Step 1

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

$\frac{lim_{x \to 1} x^{3} - 2 x^{2} - 2 x - 3}{lim_{x \to 1} x^{4} - 2 x^{3} - 27}$

Step 2

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

$\frac{lim_{x \to 1} x^{3} - lim_{x \to 1} 2 x^{2} - lim_{x \to 1} 2 x - lim_{x \to 1} 3}{lim_{x \to 1} x^{4} - 2 x^{3} - 27}$

Step 3

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

$\frac{{(lim_{x \to 1} x)}^{3} - lim_{x \to 1} 2 x^{2} - lim_{x \to 1} 2 x - lim_{x \to 1} 3}{lim_{x \to 1} x^{4} - 2 x^{3} - 27}$

Step 4

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

$\frac{{(lim_{x \to 1} x)}^{3} - 2 lim_{x \to 1} x^{2} - lim_{x \to 1} 2 x - lim_{x \to 1} 3}{lim_{x \to 1} x^{4} - 2 x^{3} - 27}$

Step 5

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

$\frac{{(lim_{x \to 1} x)}^{3} - 2 {(lim_{x \to 1} x)}^{2} - lim_{x \to 1} 2 x - lim_{x \to 1} 3}{lim_{x \to 1} x^{4} - 2 x^{3} - 27}$

Step 6

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

$\frac{{(lim_{x \to 1} x)}^{3} - 2 {(lim_{x \to 1} x)}^{2} - 2 lim_{x \to 1} x - lim_{x \to 1} 3}{lim_{x \to 1} x^{4} - 2 x^{3} - 27}$

Step 7

Evaluate the limit of $3$ which is constant as $x$ approaches $1$ .

$\frac{{(lim_{x \to 1} x)}^{3} - 2 {(lim_{x \to 1} x)}^{2} - 2 lim_{x \to 1} x - 1 \cdot 3}{lim_{x \to 1} x^{4} - 2 x^{3} - 27}$

Step 8

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

$\frac{{(lim_{x \to 1} x)}^{3} - 2 {(lim_{x \to 1} x)}^{2} - 2 lim_{x \to 1} x - 1 \cdot 3}{lim_{x \to 1} x^{4} - lim_{x \to 1} 2 x^{3} - lim_{x \to 1} 27}$

Step 9

Move the exponent $4$ from $x^{4}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 1} x)}^{3} - 2 {(lim_{x \to 1} x)}^{2} - 2 lim_{x \to 1} x - 1 \cdot 3}{{(lim_{x \to 1} x)}^{4} - lim_{x \to 1} 2 x^{3} - lim_{x \to 1} 27}$

Step 10

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

$\frac{{(lim_{x \to 1} x)}^{3} - 2 {(lim_{x \to 1} x)}^{2} - 2 lim_{x \to 1} x - 1 \cdot 3}{{(lim_{x \to 1} x)}^{4} - 2 lim_{x \to 1} x^{3} - lim_{x \to 1} 27}$

Step 11

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

$\frac{{(lim_{x \to 1} x)}^{3} - 2 {(lim_{x \to 1} x)}^{2} - 2 lim_{x \to 1} x - 1 \cdot 3}{{(lim_{x \to 1} x)}^{4} - 2 {(lim_{x \to 1} x)}^{3} - lim_{x \to 1} 27}$

Step 12

Evaluate the limit of $27$ which is constant as $x$ approaches $1$ .

$\frac{{(lim_{x \to 1} x)}^{3} - 2 {(lim_{x \to 1} x)}^{2} - 2 lim_{x \to 1} x - 1 \cdot 3}{{(lim_{x \to 1} x)}^{4} - 2 {(lim_{x \to 1} x)}^{3} - 1 \cdot 27}$

Step 13

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

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Step 13.1