Calculus Examples

$lim_{x \to 0} x^{2} + 3 x - 2 + \frac{2 x}{x - 1}$

Step 1

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

$lim_{x \to 0} x^{2} + lim_{x \to 0} 3 x - lim_{x \to 0} 2 + lim_{x \to 0} \frac{2 x}{x - 1}$

Step 2

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

${(lim_{x \to 0} x)}^{2} + lim_{x \to 0} 3 x - lim_{x \to 0} 2 + lim_{x \to 0} \frac{2 x}{x - 1}$

Step 3

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

${(lim_{x \to 0} x)}^{2} + 3 lim_{x \to 0} x - lim_{x \to 0} 2 + lim_{x \to 0} \frac{2 x}{x - 1}$

Step 4

Evaluate the limit of $2$ which is constant as $x$ approaches $0$ .

${(lim_{x \to 0} x)}^{2} + 3 lim_{x \to 0} x - 1 \cdot 2 + lim_{x \to 0} \frac{2 x}{x - 1}$

Step 5

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

${(lim_{x \to 0} x)}^{2} + 3 lim_{x \to 0} x - 1 \cdot 2 + 2 lim_{x \to 0} \frac{x}{x - 1}$

Step 6

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

${(lim_{x \to 0} x)}^{2} + 3 lim_{x \to 0} x - 1 \cdot 2 + 2 \frac{lim_{x \to 0} x}{lim_{x \to 0} x - 1}$

Step 7

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

${(lim_{x \to 0} x)}^{2} + 3 lim_{x \to 0} x - 1 \cdot 2 + 2 \frac{lim_{x \to 0} x}{lim_{x \to 0} x - lim_{x \to 0} 1}$

Step 8

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

${(lim_{x \to 0} x)}^{2} + 3 lim_{x \to 0} x - 1 \cdot 2 + 2 \frac{lim_{x \to 0} x}{lim_{x \to 0} x - 1 \cdot 1}$

Step 9

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

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Step 9.1

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

$0^{2} + 3 lim_{x \to 0} x - 1 \cdot 2 + 2 \frac{lim_{x \to 0} x}{lim_{x \to 0} x - 1 \cdot 1}$

Step 9.2

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

$0^{2} + 3 \cdot 0 - 1 \cdot 2 + 2 \frac{lim_{x \to 0} x}{lim_{x \to 0} x - 1 \cdot 1}$

Step 9.3

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

$0^{2} + 3 \cdot 0 - 1 \cdot 2 + 2 \frac{0}{lim_{x \to 0} x - 1 \cdot 1}$

Step 9.4

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

$0^{2} + 3 \cdot 0 - 1 \cdot 2 + 2 \frac{0}{0 - 1 \cdot 1}$

Step 10

Simplify the answer.

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Step 10.1

Simplify each term.

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Step 10.1.1

Raising $0$ to any positive power yields $0$ .

$0 + 3 \cdot 0 - 1 \cdot 2 + 2 \frac{0}{0 - 1 \cdot 1}$

Step 10.1.2

Multiply $3$ by $0$ .

$0 + 0 - 1 \cdot 2 + 2 \frac{0}{0 - 1 \cdot 1}$

Step 10.1.3

Multiply $- 1$ by $2$ .

$0 + 0 - 2 + 2 \frac{0}{0 - 1 \cdot 1}$

Step 10.1.4

Simplify the denominator.

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Step 10.1.4.1

Multiply $- 1$ by $1$ .

$0 + 0 - 2 + 2 \frac{0}{0 - 1}$

Step 10.1.4.2

Subtract $1$ from $0$ .

$0 + 0 - 2 + 2 (\frac{0}{- 1})$

Step 10.1.5

Divide $0$ by $- 1$ .

$0 + 0 - 2 + 2 \cdot 0$

Step 10.1.6

Multiply $2$ by $0$ .

$0 + 0 - 2 + 0$

Step 10.2

Add $0$ and $0$ .

$0 - 2 + 0$

Step 10.3

Subtract $2$ from $0$ .

$- 2 + 0$

Step 10.4

Add $- 2$ and $0$ .

$- 2$