Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

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Step 11.1

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

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

Step 11.2

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

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

Step 11.3

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

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

Step 12

Simplify the answer.

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Step 12.1

Simplify the numerator.

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Step 12.1.1

Multiply $- 1$ by $3$ .

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

Step 12.1.2

Subtract $3$ from $1$ .

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

Step 12.1.3

One to any power is one.

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

Step 12.1.4

Multiply $- 1$ by $2$ .

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

Step 12.1.5

Subtract $2$ from $1$ .

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

Step 12.2

Simplify the denominator.

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Step 12.2.1

One to any power is one.

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

Step 12.2.2

Add $1$ and $1$ .

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

Step 12.3

Multiply $- 2$ by $- 1$ .

$\frac{2}{2}$

Step 12.4

Divide $2$ by $2$ .

$1$