Calculus Examples

$lim_{h \to 0} \frac{17 x + h}{x^{3} (x - h)}$

Step 1

Move the term $\frac{1}{x^{3}}$ outside of the limit because it is constant with respect to $h$ .

$\frac{1}{x^{3}} lim_{h \to 0} \frac{17 x + h}{x - h}$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $h$ approaches $0$ .

$\frac{1}{x^{3}} \cdot \frac{lim_{h \to 0} 17 x + h}{lim_{h \to 0} x - h}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$\frac{1}{x^{3}} \cdot \frac{lim_{h \to 0} 17 x + lim_{h \to 0} h}{lim_{h \to 0} x - h}$

Step 4

Evaluate the limit of $17 x$ which is constant as $h$ approaches $0$ .

$\frac{1}{x^{3}} \cdot \frac{17 x + lim_{h \to 0} h}{lim_{h \to 0} x - h}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$\frac{1}{x^{3}} \cdot \frac{17 x + lim_{h \to 0} h}{lim_{h \to 0} x - lim_{h \to 0} h}$

Step 6

Evaluate the limit of $x$ which is constant as $h$ approaches $0$ .

$\frac{1}{x^{3}} \cdot \frac{17 x + lim_{h \to 0} h}{x - lim_{h \to 0} h}$

Step 7

Evaluate the limits by plugging in $0$ for all occurrences of $h$ .

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Step 7.1

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$\frac{1}{x^{3}} \cdot \frac{17 x + 0}{x - lim_{h \to 0} h}$

Step 7.2

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$\frac{1}{x^{3}} \cdot \frac{17 x + 0}{x + 0}$

Step 8

Simplify the answer.

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Step 8.1

Add $17 x$ and $0$ .

$\frac{1}{x^{3}} \cdot \frac{17 x}{x + 0}$

Step 8.2

Add $x$ and $0$ .

$\frac{1}{x^{3}} \cdot \frac{17 x}{x}$

Step 8.3

Combine.

$\frac{1 (17 x)}{x^{3} x}$

Step 8.4

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

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Step 8.4.1

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

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Step 8.4.1.1

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

$\frac{1 (17 x)}{x^{3} x^{1}}$

Step 8.4.1.2

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

$\frac{1 (17 x)}{x^{3 + 1}}$

Step 8.4.2

Add $3$ and $1$ .

$\frac{1 (17 x)}{x^{4}}$

Step 8.5

Multiply $17 x$ by $1$ .

$\frac{17 x}{x^{4}}$

Step 8.6

Cancel the common factor of $x$ and $x^{4}$ .

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Step 8.6.1

Factor $x$ out of $17 x$ .

$\frac{x \cdot 17}{x^{4}}$

Step 8.6.2

Cancel the common factors.

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Step 8.6.2.1

Factor $x$ out of $x^{4}$ .

$\frac{x \cdot 17}{x \cdot x^{3}}$

Step 8.6.2.2

Cancel the common factor.

$\frac{x \cdot 17}{x \cdot x^{3}}$

Step 8.6.2.3

Rewrite the expression.

$\frac{17}{x^{3}}$