Calculus Examples

$lim_{x \to 2} \frac{x^{3} - x^{2} - x - 2}{2 x^{3} - 5 y^{2} + 5 y - 6}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $2$ .

$\frac{lim_{x \to 2} x^{3} - x^{2} - x - 2}{lim_{x \to 2} 2 x^{3} - 5 y^{2} + 5 y - 6}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{lim_{x \to 2} x^{3} - lim_{x \to 2} x^{2} - lim_{x \to 2} x - lim_{x \to 2} 2}{lim_{x \to 2} 2 x^{3} - 5 y^{2} + 5 y - 6}$

Step 3

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

$\frac{{(lim_{x \to 2} x)}^{3} - lim_{x \to 2} x^{2} - lim_{x \to 2} x - lim_{x \to 2} 2}{lim_{x \to 2} 2 x^{3} - 5 y^{2} + 5 y - 6}$

Step 4

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

$\frac{{(lim_{x \to 2} x)}^{3} - {(lim_{x \to 2} x)}^{2} - lim_{x \to 2} x - lim_{x \to 2} 2}{lim_{x \to 2} 2 x^{3} - 5 y^{2} + 5 y - 6}$

Step 5

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

$\frac{{(lim_{x \to 2} x)}^{3} - {(lim_{x \to 2} x)}^{2} - lim_{x \to 2} x - 1 \cdot 2}{lim_{x \to 2} 2 x^{3} - 5 y^{2} + 5 y - 6}$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{{(lim_{x \to 2} x)}^{3} - {(lim_{x \to 2} x)}^{2} - lim_{x \to 2} x - 1 \cdot 2}{lim_{x \to 2} 2 x^{3} - lim_{x \to 2} 5 y^{2} + lim_{x \to 2} 5 y - lim_{x \to 2} 6}$

Step 7

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

$\frac{{(lim_{x \to 2} x)}^{3} - {(lim_{x \to 2} x)}^{2} - lim_{x \to 2} x - 1 \cdot 2}{2 lim_{x \to 2} x^{3} - lim_{x \to 2} 5 y^{2} + lim_{x \to 2} 5 y - lim_{x \to 2} 6}$

Step 8

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

$\frac{{(lim_{x \to 2} x)}^{3} - {(lim_{x \to 2} x)}^{2} - lim_{x \to 2} x - 1 \cdot 2}{2 {(lim_{x \to 2} x)}^{3} - lim_{x \to 2} 5 y^{2} + lim_{x \to 2} 5 y - lim_{x \to 2} 6}$

Step 9

Evaluate the limit of $5 y^{2}$ which is constant as $x$ approaches $2$ .

$\frac{{(lim_{x \to 2} x)}^{3} - {(lim_{x \to 2} x)}^{2} - lim_{x \to 2} x - 1 \cdot 2}{2 {(lim_{x \to 2} x)}^{3} - 5 y^{2} + lim_{x \to 2} 5 y - lim_{x \to 2} 6}$

Step 10

Evaluate the limit of $5 y$ which is constant as $x$ approaches $2$ .

$\frac{{(lim_{x \to 2} x)}^{3} - {(lim_{x \to 2} x)}^{2} - lim_{x \to 2} x - 1 \cdot 2}{2 {(lim_{x \to 2} x)}^{3} - 5 y^{2} + 5 y - lim_{x \to 2} 6}$

Step 11

Evaluate the limit of $6$ which is constant as $x$ approaches $2$ .

$\frac{{(lim_{x \to 2} x)}^{3} - {(lim_{x \to 2} x)}^{2} - lim_{x \to 2} x - 1 \cdot 2}{2 {(lim_{x \to 2} x)}^{3} - 5 y^{2} + 5 y - 1 \cdot 6}$

Step 12

Evaluate the limits by plugging in $2$ for all occurrences of $x$ .

Tap for more steps...

Step 12.1

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{2^{3} - {(lim_{x \to 2} x)}^{2} - lim_{x \to 2} x - 1 \cdot 2}{2 {(lim_{x \to 2} x)}^{3} - 5 y^{2} + 5 y - 1 \cdot 6}$

Step 12.2

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{2^{3} - 2^{2} - lim_{x \to 2} x - 1 \cdot 2}{2 {(lim_{x \to 2} x)}^{3} - 5 y^{2} + 5 y - 1 \cdot 6}$

Step 12.3

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{2^{3} - 2^{2} - 2 - 1 \cdot 2}{2 {(lim_{x \to 2} x)}^{3} - 5 y^{2} + 5 y - 1 \cdot 6}$

Step 12.4

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{2^{3} - 2^{2} - 2 - 1 \cdot 2}{2 \cdot 2^{3} - 5 y^{2} + 5 y - 1 \cdot 6}$

Step 13

Simplify the answer.

Tap for more steps...

Step 13.1

Simplify the numerator.

Tap for more steps...

Step 13.1.1

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

$\frac{8 - 2^{2} - 2 - 1 \cdot 2}{2 \cdot 2^{3} - 5 y^{2} + 5 y - 1 \cdot 6}$

Step 13.1.2

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

$\frac{8 - 1 \cdot 4 - 2 - 1 \cdot 2}{2 \cdot 2^{3} - 5 y^{2} + 5 y - 1 \cdot 6}$

Step 13.1.3

Multiply $- 1$ by $4$ .

$\frac{8 - 4 - 2 - 1 \cdot 2}{2 \cdot 2^{3} - 5 y^{2} + 5 y - 1 \cdot 6}$

Step 13.1.4

Multiply $- 1$ by $2$ .

$\frac{8 - 4 - 2 - 2}{2 \cdot 2^{3} - 5 y^{2} + 5 y - 1 \cdot 6}$

Step 13.1.5

Subtract $4$ from $8$ .

$\frac{4 - 2 - 2}{2 \cdot 2^{3} - 5 y^{2} + 5 y - 1 \cdot 6}$

Step 13.1.6

Subtract $2$ from $4$ .

$\frac{2 - 2}{2 \cdot 2^{3} - 5 y^{2} + 5 y - 1 \cdot 6}$

Step 13.1.7

Subtract $2$ from $2$ .

$\frac{0}{2 \cdot 2^{3} - 5 y^{2} + 5 y - 1 \cdot 6}$

Step 13.2

Simplify the denominator.

Tap for more steps...

Step 13.2.1

Multiply $2$ by $2^{3}$ by adding the exponents.

Tap for more steps...

Step 13.2.1.1

Multiply $2$ by $2^{3}$ .

Tap for more steps...

Step 13.2.1.1.1

Raise $2$ to the power of $1$ .

$\frac{0}{2^{1} \cdot 2^{3} - 5 y^{2} + 5 y - 1 \cdot 6}$

Step 13.2.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{0}{2^{1 + 3} - 5 y^{2} + 5 y - 1 \cdot 6}$

Step 13.2.1.2

Add $1$ and $3$ .

$\frac{0}{2^{4} - 5 y^{2} + 5 y - 1 \cdot 6}$

Step 13.2.2

Raise $2$ to the power of $4$ .

$\frac{0}{16 - 5 y^{2} + 5 y - 1 \cdot 6}$

Step 13.2.3

Multiply $- 1$ by $6$ .

$\frac{0}{16 - 5 y^{2} + 5 y - 6}$

Step 13.2.4

Subtract $6$ from $16$ .

$\frac{0}{- 5 y^{2} + 5 y + 10}$

Step 13.2.5

Rewrite $- 5 y^{2} + 5 y + 10$ in a factored form.

Tap for more steps...

Step 13.2.5.1

Factor $5$ out of $- 5 y^{2} + 5 y + 10$ .

Tap for more steps...

Step 13.2.5.1.1

Factor $5$ out of $- 5 y^{2}$ .

$\frac{0}{5 (- y^{2}) + 5 y + 10}$

Step 13.2.5.1.2

Factor $5$ out of $5 y$ .

$\frac{0}{5 (- y^{2}) + 5 (y) + 10}$

Step 13.2.5.1.3

Factor $5$ out of $10$ .

$\frac{0}{5 (- y^{2}) + 5 y + 5 \cdot 2}$

Step 13.2.5.1.4

Factor $5$ out of $5 (- y^{2}) + 5 y$ .

$\frac{0}{5 (- y^{2} + y) + 5 \cdot 2}$

Step 13.2.5.1.5

Factor $5$ out of $5 (- y^{2} + y) + 5 \cdot 2$ .

$\frac{0}{5 (- y^{2} + y + 2)}$

Step 13.2.5.2

Factor by grouping.

Tap for more steps...

Step 13.2.5.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 2 = - 2$ and whose sum is $b = 1$ .

Tap for more steps...

Step 13.2.5.2.1.1

Multiply by $1$ .

$\frac{0}{5 (- y^{2} + 1 y + 2)}$

Step 13.2.5.2.1.2

Rewrite $1$ as $- 1$ plus $2$

$\frac{0}{5 (- y^{2} + (- 1 + 2) y + 2)}$

Step 13.2.5.2.1.3

Apply the distributive property.

$\frac{0}{5 (- y^{2} - 1 y + 2 y + 2)}$

Step 13.2.5.2.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 13.2.5.2.2.1

Group the first two terms and the last two terms.

$\frac{0}{5 ((- y^{2} - 1 y) + 2 y + 2)}$

Step 13.2.5.2.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{0}{5 (y (- y - 1) - 2 (- y - 1))}$

Step 13.2.5.2.3

Factor the polynomial by factoring out the greatest common factor, $- y - 1$ .

$\frac{0}{5 ((- y - 1) (y - 2))}$

$\frac{0}{5 (- y - 1) (y - 2)}$

Step 13.3

Cancel the common factor of $0$ and $5$ .

Tap for more steps...

Step 13.3.1

Factor $5$ out of $0$ .

$\frac{5 (0)}{5 (- y - 1) (y - 2)}$

Step 13.3.2

Cancel the common factors.

Tap for more steps...

Step 13.3.2.1

Factor $5$ out of $5 (- y - 1) (y - 2)$ .

$\frac{5 (0)}{5 ((- y - 1) (y - 2))}$

Step 13.3.2.2

Cancel the common factor.

$\frac{5 \cdot 0}{5 ((- y - 1) (y - 2))}$

Step 13.3.2.3

Rewrite the expression.

$\frac{0}{(- y - 1) (y - 2)}$

Step 13.4

Divide $0$ by $(- y - 1) (y - 2)$ .

$0$