Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Evaluate the limit of $3$ which is constant as $x$ approaches $1$ .

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

Step 7

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

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

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

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

One to any power is one.

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

Step 8.1.2

Multiply $- 1$ by $2$ .

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

Step 8.1.3

Add $1$ and $1$ .

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

Step 8.1.4

Subtract $2$ from $2$ .

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

Step 8.2

Simplify the denominator.

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Step 8.2.1

Multiply $- 1$ by $3$ .

$\frac{0}{1 - 3}$

Step 8.2.2

Subtract $3$ from $1$ .

$\frac{0}{- 2}$

Step 8.3

Divide $0$ by $- 2$ .

$0$