Calculus Examples

$\frac{lim_{x \to 2} 3 x^{2} - 7 x + 2}{2 - x}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{lim_{x \to 2} 3 x^{2} - lim_{x \to 2} 7 x + lim_{x \to 2} 2}{2 - x}$

Step 2

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

$\frac{3 lim_{x \to 2} x^{2} - lim_{x \to 2} 7 x + lim_{x \to 2} 2}{2 - x}$

Step 3

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

$\frac{3 {(lim_{x \to 2} x)}^{2} - lim_{x \to 2} 7 x + lim_{x \to 2} 2}{2 - x}$

Step 4

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$\frac{3 {(lim_{x \to 2} x)}^{2} - 7 lim_{x \to 2} x + lim_{x \to 2} 2}{2 - x}$

Step 5

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

$\frac{3 {(lim_{x \to 2} x)}^{2} - 7 lim_{x \to 2} x + 2}{2 - x}$

Step 6

Evaluate the limits by plugging in $2$ for all occurrences of $x$ .

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Step 6.1

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{3 \cdot 2^{2} - 7 lim_{x \to 2} x + 2}{2 - x}$

Step 6.2

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{3 \cdot 2^{2} - 7 \cdot 2 + 2}{2 - x}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $2$ to the power of $2$ .

$\frac{3 \cdot 4 - 7 \cdot 2 + 2}{2 - x}$

Step 7.1.2

Multiply $3$ by $4$ .

$\frac{12 - 7 \cdot 2 + 2}{2 - x}$

Step 7.1.3

Multiply $- 7$ by $2$ .

$\frac{12 - 14 + 2}{2 - x}$

Step 7.1.4

Subtract $14$ from $12$ .

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

Step 7.1.5

Add $- 2$ and $2$ .

$\frac{0}{2 - x}$

Step 7.2

Divide $0$ by $2 - x$ .

$0$