Calculus Examples

$\frac{lim_{x \to 1} x^{19} + x^{8} - 2}{x - 1}$

Step 1

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

$\frac{lim_{x \to 1} x^{19} + lim_{x \to 1} x^{8} - lim_{x \to 1} 2}{x - 1}$

Step 2

Move the exponent $19$ from $x^{19}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 1} x)}^{19} + lim_{x \to 1} x^{8} - lim_{x \to 1} 2}{x - 1}$

Step 3

Move the exponent $8$ from $x^{8}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 1} x)}^{19} + {(lim_{x \to 1} x)}^{8} - lim_{x \to 1} 2}{x - 1}$

Step 4

Evaluate the limit of $2$ which is constant as $x$ approaches $1$ .

$\frac{{(lim_{x \to 1} x)}^{19} + {(lim_{x \to 1} x)}^{8} - 1 \cdot 2}{x - 1}$

Step 5

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

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Step 5.1

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

$\frac{1^{19} + {(lim_{x \to 1} x)}^{8} - 1 \cdot 2}{x - 1}$

Step 5.2

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

$\frac{1^{19} + 1^{8} - 1 \cdot 2}{x - 1}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

One to any power is one.

$\frac{1 + 1^{8} - 1 \cdot 2}{x - 1}$

Step 6.1.2

One to any power is one.

$\frac{1 + 1 - 1 \cdot 2}{x - 1}$

Step 6.1.3

Multiply $- 1$ by $2$ .

$\frac{1 + 1 - 2}{x - 1}$

Step 6.1.4

Add $1$ and $1$ .

$\frac{2 - 2}{x - 1}$

Step 6.1.5

Subtract $2$ from $2$ .

$\frac{0}{x - 1}$

Step 6.2

Divide $0$ by $x - 1$ .

$0$