Calculus Examples

$lim_{x \to 8} z + 1 - (\frac{z^{3} - 5}{z^{2} - 3})$

Step 1

Evaluate the limit of $z + 1 - (\frac{z^{3} - 5}{z^{2} - 3})$ which is constant as $x$ approaches $8$ .

$z + 1 - (\frac{z^{3} - 5}{z^{2} - 3})$

Step 2

Simplify the answer.

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Step 2.1

To write $z$ as a fraction with a common denominator, multiply by $\frac{z^{2} - 3}{z^{2} - 3}$ .

$\frac{z (z^{2} - 3)}{z^{2} - 3} - \frac{z^{3} - 5}{z^{2} - 3} + 1$

Step 2.2

Combine the numerators over the common denominator.

$\frac{z (z^{2} - 3) - (z^{3} - 5)}{z^{2} - 3} + 1$

Step 2.3

Simplify the numerator.

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Step 2.3.1

Apply the distributive property.

$\frac{z \cdot z^{2} + z \cdot - 3 - (z^{3} - 5)}{z^{2} - 3} + 1$

Step 2.3.2

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

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Step 2.3.2.1

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

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Step 2.3.2.1.1

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

$\frac{z^{1} z^{2} + z \cdot - 3 - (z^{3} - 5)}{z^{2} - 3} + 1$

Step 2.3.2.1.2

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

$\frac{z^{1 + 2} + z \cdot - 3 - (z^{3} - 5)}{z^{2} - 3} + 1$

Step 2.3.2.2

Add $1$ and $2$ .

$\frac{z^{3} + z \cdot - 3 - (z^{3} - 5)}{z^{2} - 3} + 1$

Step 2.3.3

Move $- 3$ to the left of $z$ .

$\frac{z^{3} - 3 \cdot z - (z^{3} - 5)}{z^{2} - 3} + 1$

Step 2.3.4

Apply the distributive property.

$\frac{z^{3} - 3 z - z^{3} - - 5}{z^{2} - 3} + 1$

Step 2.3.5

Multiply $- 1$ by $- 5$ .

$\frac{z^{3} - 3 z - z^{3} + 5}{z^{2} - 3} + 1$

Step 2.3.6

Subtract $z^{3}$ from $z^{3}$ .

$\frac{- 3 z + 0 + 5}{z^{2} - 3} + 1$

Step 2.3.7

Add $- 3 z$ and $0$ .

$\frac{- 3 z + 5}{z^{2} - 3} + 1$

Step 2.4

Write $1$ as a fraction with a common denominator.

$\frac{- 3 z + 5}{z^{2} - 3} + \frac{z^{2} - 3}{z^{2} - 3}$

Step 2.5

Combine the numerators over the common denominator.

$\frac{- 3 z + 5 + z^{2} - 3}{z^{2} - 3}$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Subtract $3$ from $5$ .

$\frac{- 3 z + z^{2} + 2}{z^{2} - 3}$

Step 2.6.2

Let $u = z$ . Substitute $u$ for all occurrences of $z$ .

$\frac{- 3 u + u^{2} + 2}{z^{2} - 3}$

Step 2.6.3

Factor $- 3 u + u^{2} + 2$ using the AC method.

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Step 2.6.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $2$ and whose sum is $- 3$ .

$- 2, - 1$

Step 2.6.3.2

Write the factored form using these integers.

$\frac{(u - 2) (u - 1)}{z^{2} - 3}$

Step 2.6.4

Replace all occurrences of $u$ with $z$ .

$\frac{(z - 2) (z - 1)}{z^{2} - 3}$