Calculus Examples

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

Step 1

Change the two-sided limit into a left sided limit.

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

Step 2

Evaluate the limits by plugging in the value for the variable.

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Step 2.1

Evaluate the limit of $\frac{10 x - 1}{| 10 x^{3} - x^{2} |}$ by plugging in $0.1$ for $x$ .

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

Step 2.2

Simplify each term.

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Step 2.2.1

Raise $0.1$ to the power of $3$ .

$\frac{10 \cdot 0.1 - 1}{| 10 \cdot 0.001 - {0.1}^{2} |}$

Step 2.2.2

Multiply $10$ by $0.001$ .

$\frac{10 \cdot 0.1 - 1}{| 0.01 - {0.1}^{2} |}$

Step 2.2.3

Raise $0.1$ to the power of $2$ .

$\frac{10 \cdot 0.1 - 1}{| 0.01 - 1 \cdot 0.01 |}$

Step 2.2.4

Multiply $- 1$ by $0.01$ .

$\frac{10 \cdot 0.1 - 1}{| 0.01 - 0.01 |}$

Step 2.3

Subtract $0.01$ from $0.01$ .

$\frac{10 \cdot 0.1 - 1}{| 0 |}$

Step 2.4

The absolute value is the distance between a number and zero. The distance between $0$ and $0$ is $0$ .

$\frac{10 \cdot 0.1 - 1}{0}$

Step 2.5

Since $\frac{10 \cdot 0.1 - 1}{0}$ is undefined, the limit does not exist.

$Does not exist$