Calculus Examples

$\frac{lim_{x \to 0.1} 10 x - 1}{| 10 x^{3} - x^{2} |}$

Step 1

Evaluate the limit.

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Step 1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0.1$ .

$\frac{lim_{x \to 0.1} 10 x - lim_{x \to 0.1} 1}{| 10 x^{3} - x^{2} |}$

Step 1.2

Move the term $10$ outside of the limit because it is constant with respect to $x$ .

$\frac{10 lim_{x \to 0.1} x - lim_{x \to 0.1} 1}{| 10 x^{3} - x^{2} |}$

Step 1.3

Evaluate the limit of $1$ which is constant as $x$ approaches $0.1$ .

$\frac{10 lim_{x \to 0.1} x - 1 \cdot 1}{| 10 x^{3} - x^{2} |}$

Step 2

Evaluate the limit of $x$ by plugging in $0.1$ for $x$ .

$\frac{10 \cdot 0.1 - 1 \cdot 1}{| 10 x^{3} - x^{2} |}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Multiply $10$ by $0.1$ .

$\frac{1 - 1 \cdot 1}{| 10 x^{3} - x^{2} |}$

Step 3.1.2

Multiply $- 1$ by $1$ .

$\frac{1 - 1}{| 10 x^{3} - x^{2} |}$

Step 3.1.3

Subtract $1$ from $1$ .

$\frac{0}{| 10 x^{3} - x^{2} |}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Factor $x^{2}$ out of $10 x^{3} - x^{2}$ .

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Step 3.2.1.1

Factor $x^{2}$ out of $10 x^{3}$ .

$\frac{0}{| x^{2} (10 x) - x^{2} |}$

Step 3.2.1.2

Factor $x^{2}$ out of $- x^{2}$ .

$\frac{0}{| x^{2} (10 x) + x^{2} \cdot - 1 |}$

Step 3.2.1.3

Factor $x^{2}$ out of $x^{2} (10 x) + x^{2} \cdot - 1$ .

$\frac{0}{| x^{2} (10 x - 1) |}$

Step 3.2.2

Apply the distributive property.

$\frac{0}{| x^{2} (10 x) + x^{2} \cdot - 1 |}$

Step 3.2.3

Rewrite using the commutative property of multiplication.

$\frac{0}{| 10 x^{2} x + x^{2} \cdot - 1 |}$

Step 3.2.4

Move $- 1$ to the left of $x^{2}$ .

$\frac{0}{| 10 x^{2} x - 1 \cdot x^{2} |}$

Step 3.2.5

Simplify each term.

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Step 3.2.5.1

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

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Step 3.2.5.1.1

Move $x$ .

$\frac{0}{| 10 (x \cdot x^{2}) - 1 \cdot x^{2} |}$

Step 3.2.5.1.2

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

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Step 3.2.5.1.2.1

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

$\frac{0}{| 10 (x^{1} x^{2}) - 1 \cdot x^{2} |}$

Step 3.2.5.1.2.2

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

$\frac{0}{| 10 x^{1 + 2} - 1 \cdot x^{2} |}$

Step 3.2.5.1.3

Add $1$ and $2$ .

$\frac{0}{| 10 x^{3} - 1 \cdot x^{2} |}$

Step 3.2.5.2

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$\frac{0}{| 10 x^{3} - x^{2} |}$

Step 3.2.6

Factor $x^{2}$ out of $10 x^{3} - x^{2}$ .

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Step 3.2.6.1

Factor $x^{2}$ out of $10 x^{3}$ .

$\frac{0}{| x^{2} (10 x) - x^{2} |}$

Step 3.2.6.2

Factor $x^{2}$ out of $- x^{2}$ .

$\frac{0}{| x^{2} (10 x) + x^{2} \cdot - 1 |}$

Step 3.2.6.3

Factor $x^{2}$ out of $x^{2} (10 x) + x^{2} \cdot - 1$ .

$\frac{0}{| x^{2} (10 x - 1) |}$

Step 3.3

Divide $0$ by $| x^{2} (10 x - 1) |$ .

$0$