Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

Split the limit using the Sum of Limits Rule on the limit as

$x$ approaches

$1$ .

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

Step 1.2

Move the term

$2$ outside of the limit because it is constant with respect to

$x$ .

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

Step 1.3

Move the exponent

$2$ from

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

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

Step 1.4

Evaluate the limit of

$2$ which is constant as

$x$ approaches

$1$ .

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

Step 2

Evaluate the limit of

$x$ by plugging in

$1$ for

$x$ .

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

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

One to any power is one.

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

Step 3.1.2

Multiply

$2$ by

$1$ .

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

Step 3.1.3

Multiply

$- 1$ by

$2$ .

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

Step 3.1.4

Subtract

$2$ from

$2$ .

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

Step 3.2

Factor

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

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Step 3.2.1

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

$2$ .

$- 1, 3$

Step 3.2.2

Write the factored form using these integers.

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

Step 3.3

Divide

$0$ by

$(x - 1) (x + 3)$ .

$0$