Calculus Examples

$lim_{x \to 2} - \frac{x^{2}}{2} - x + \frac{5}{2}$

Step 1

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

$- lim_{x \to 2} \frac{x^{2}}{2} - lim_{x \to 2} x + lim_{x \to 2} \frac{5}{2}$

Step 2

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

$- \frac{1}{2} lim_{x \to 2} x^{2} - lim_{x \to 2} x + lim_{x \to 2} \frac{5}{2}$

Step 3

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

$- \frac{1}{2} {(lim_{x \to 2} x)}^{2} - lim_{x \to 2} x + lim_{x \to 2} \frac{5}{2}$

Step 4

Evaluate the limit of $\frac{5}{2}$ which is constant as $x$ approaches $2$ .

$- \frac{1}{2} {(lim_{x \to 2} x)}^{2} - lim_{x \to 2} x + \frac{5}{2}$

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 $x$ .

$- \frac{1}{2} \cdot 2^{2} - lim_{x \to 2} x + \frac{5}{2}$

Step 5.2

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

$- \frac{1}{2} \cdot 2^{2} - 2 + \frac{5}{2}$

Step 6

Simplify the answer.

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Step 6.1

Simplify each term.

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Step 6.1.1

Cancel the common factor of $2$ .

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Step 6.1.1.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$\frac{- 1}{2} \cdot 2^{2} - 2 + \frac{5}{2}$

Step 6.1.1.2

Factor $2$ out of $2^{2}$ .

$\frac{- 1}{2} \cdot (2 \cdot 2) - 2 + \frac{5}{2}$

Step 6.1.1.3

Cancel the common factor.

$\frac{- 1}{2} \cdot (2 \cdot 2) - 2 + \frac{5}{2}$

Step 6.1.1.4

Rewrite the expression.

$- 1 \cdot 2 - 2 + \frac{5}{2}$

Step 6.1.2

Multiply $- 1$ by $2$ .

$- 2 - 2 + \frac{5}{2}$

Step 6.2

Find the common denominator.

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Step 6.2.1

Write $- 2$ as a fraction with denominator $1$ .

$\frac{- 2}{1} - 2 + \frac{5}{2}$

Step 6.2.2

Multiply $\frac{- 2}{1}$ by $\frac{2}{2}$ .

$\frac{- 2}{1} \cdot \frac{2}{2} - 2 + \frac{5}{2}$

Step 6.2.3

Multiply $\frac{- 2}{1}$ by $\frac{2}{2}$ .

$\frac{- 2 \cdot 2}{2} - 2 + \frac{5}{2}$

Step 6.2.4

Write $- 2$ as a fraction with denominator $1$ .

$\frac{- 2 \cdot 2}{2} + \frac{- 2}{1} + \frac{5}{2}$

Step 6.2.5

Multiply $\frac{- 2}{1}$ by $\frac{2}{2}$ .

$\frac{- 2 \cdot 2}{2} + \frac{- 2}{1} \cdot \frac{2}{2} + \frac{5}{2}$

Step 6.2.6

Multiply $\frac{- 2}{1}$ by $\frac{2}{2}$ .

$\frac{- 2 \cdot 2}{2} + \frac{- 2 \cdot 2}{2} + \frac{5}{2}$

Step 6.3

Combine the numerators over the common denominator.

$\frac{- 2 \cdot 2 - 2 \cdot 2 + 5}{2}$

Step 6.4

Simplify each term.

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Step 6.4.1

Multiply $- 2$ by $2$ .

$\frac{- 4 - 2 \cdot 2 + 5}{2}$

Step 6.4.2

Multiply $- 2$ by $2$ .

$\frac{- 4 - 4 + 5}{2}$

Step 6.5

Subtract $4$ from $- 4$ .

$\frac{- 8 + 5}{2}$

Step 6.6

Add $- 8$ and $5$ .

$\frac{- 3}{2}$

Step 6.7

Move the negative in front of the fraction.

$- \frac{3}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \frac{3}{2}$

Decimal Form:

$- 1.5$