Calculus Examples

$\frac{lim_{x \to 1} x^{2} + x - 2}{x^{2} - 4 x + 3}$

Step 1

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

$\frac{lim_{x \to 1} x^{2} + lim_{x \to 1} x - lim_{x \to 1} 2}{x^{2} - 4 x + 3}$

Step 2

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

$\frac{{(lim_{x \to 1} x)}^{2} + lim_{x \to 1} x - lim_{x \to 1} 2}{x^{2} - 4 x + 3}$

Step 3

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

$\frac{{(lim_{x \to 1} x)}^{2} + lim_{x \to 1} x - 1 \cdot 2}{x^{2} - 4 x + 3}$

Step 4

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

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Step 4.1

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

$\frac{1^{2} + lim_{x \to 1} x - 1 \cdot 2}{x^{2} - 4 x + 3}$

Step 4.2

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

$\frac{1^{2} + 1 - 1 \cdot 2}{x^{2} - 4 x + 3}$

Step 5

Simplify the answer.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

One to any power is one.

$\frac{1 + 1 - 1 \cdot 2}{x^{2} - 4 x + 3}$

Step 5.1.2

Multiply $- 1$ by $2$ .

$\frac{1 + 1 - 2}{x^{2} - 4 x + 3}$

Step 5.1.3

Add $1$ and $1$ .

$\frac{2 - 2}{x^{2} - 4 x + 3}$

Step 5.1.4

Subtract $2$ from $2$ .

$\frac{0}{x^{2} - 4 x + 3}$

Step 5.2

Factor $x^{2} - 4 x + 3$ using the AC method.

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Step 5.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $3$ and whose sum is $- 4$ .

$- 3, - 1$

Step 5.2.2

Write the factored form using these integers.

$\frac{0}{(x - 3) (x - 1)}$

Step 5.3

Divide $0$ by $(x - 3) (x - 1)$ .

$0$