Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

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

Step 4

Evaluate the limit of $3$ which is constant as $x$ approaches $2$ .

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

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

Step 6.1.2

Multiply $- 4$ by $2$ .

$\frac{4 - 8 + 3}{{(x - 2)}^{2}}$

Step 6.1.3

Subtract $8$ from $4$ .

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

Step 6.1.4

Add $- 4$ and $3$ .

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

Step 6.2

Move the negative in front of the fraction.

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