Calculus Examples

$lim_{x \to 1} \frac{100 x^{99} - 2 x + 1}{10 x}$

Step 1

Move the term $\frac{1}{10}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{10} lim_{x \to 1} \frac{100 x^{99} - 2 x + 1}{x}$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $1$ .

$\frac{1}{10} \cdot \frac{lim_{x \to 1} 100 x^{99} - 2 x + 1}{lim_{x \to 1} x}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$\frac{1}{10} \cdot \frac{lim_{x \to 1} 100 x^{99} - lim_{x \to 1} 2 x + lim_{x \to 1} 1}{lim_{x \to 1} x}$

Step 4

Move the term $100$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{10} \cdot \frac{100 lim_{x \to 1} x^{99} - lim_{x \to 1} 2 x + lim_{x \to 1} 1}{lim_{x \to 1} x}$

Step 5

Move the exponent $99$ from $x^{99}$ outside the limit using the Limits Power Rule.

$\frac{1}{10} \cdot \frac{100 {(lim_{x \to 1} x)}^{99} - lim_{x \to 1} 2 x + lim_{x \to 1} 1}{lim_{x \to 1} x}$

Step 6

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

$\frac{1}{10} \cdot \frac{100 {(lim_{x \to 1} x)}^{99} - 2 lim_{x \to 1} x + lim_{x \to 1} 1}{lim_{x \to 1} x}$

Step 7

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

$\frac{1}{10} \cdot \frac{100 {(lim_{x \to 1} x)}^{99} - 2 lim_{x \to 1} x + 1}{lim_{x \to 1} x}$

Step 8

Evaluate the limits by plugging in $1$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{1}{10} \cdot \frac{100 \cdot 1^{99} - 2 lim_{x \to 1} x + 1}{lim_{x \to 1} x}$

Step 8.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{1}{10} \cdot \frac{100 \cdot 1^{99} - 2 \cdot 1 + 1}{lim_{x \to 1} x}$

Step 8.3

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{1}{10} \cdot \frac{100 \cdot 1^{99} - 2 \cdot 1 + 1}{1}$

Step 9

Simplify the answer.

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Step 9.1

Divide $100 \cdot 1^{99} - 2 \cdot 1 + 1$ by $1$ .

$\frac{1}{10} (100 \cdot 1^{99} - 2 \cdot 1 + 1)$

Step 9.2

Simplify each term.

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Step 9.2.1

One to any power is one.

$\frac{1}{10} (100 \cdot 1 - 2 \cdot 1 + 1)$

Step 9.2.2

Multiply $100$ by $1$ .

$\frac{1}{10} (100 - 2 \cdot 1 + 1)$

Step 9.2.3

Multiply $- 2$ by $1$ .

$\frac{1}{10} (100 - 2 + 1)$

Step 9.3

Subtract $2$ from $100$ .

$\frac{1}{10} (98 + 1)$

Step 9.4

Add $98$ and $1$ .

$\frac{1}{10} \cdot 99$

Step 9.5

Combine $\frac{1}{10}$ and $99$ .

$\frac{99}{10}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{99}{10}$

Decimal Form:

$9.9$