Calculus Examples

$lim_{x \to \frac{1}{2}} \frac{2 x - 5}{7}$

Step 1

Evaluate the limit.

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Step 1.1

Move the term $\frac{1}{7}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{7} lim_{x \to \frac{1}{2}} 2 x - 5$

Step 1.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{1}{2}$ .

$\frac{1}{7} (lim_{x \to \frac{1}{2}} 2 x - lim_{x \to \frac{1}{2}} 5)$

Step 1.3

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{7} (2 lim_{x \to \frac{1}{2}} x - lim_{x \to \frac{1}{2}} 5)$

Step 1.4

Evaluate the limit of $5$ which is constant as $x$ approaches $\frac{1}{2}$ .

$\frac{1}{7} (2 lim_{x \to \frac{1}{2}} x - 1 \cdot 5)$

Step 2

Evaluate the limit of $x$ by plugging in $\frac{1}{2}$ for $x$ .

$\frac{1}{7} (2 (\frac{1}{2}) - 1 \cdot 5)$

Step 3

Simplify the answer.

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Step 3.1

Simplify each term.

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Step 3.1.1

Cancel the common factor of $2$ .

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Step 3.1.1.1

Cancel the common factor.

$\frac{1}{7} (2 (\frac{1}{2}) - 1 \cdot 5)$

Step 3.1.1.2

Rewrite the expression.

$\frac{1}{7} (1 - 1 \cdot 5)$

Step 3.1.2

Multiply $- 1$ by $5$ .

$\frac{1}{7} (1 - 5)$

Step 3.2

Subtract $5$ from $1$ .

$\frac{1}{7} \cdot - 4$

Step 3.3

Combine $\frac{1}{7}$ and $- 4$ .

$\frac{- 4}{7}$

Step 3.4

Move the negative in front of the fraction.

$- \frac{4}{7}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$- \frac{4}{7}$

Decimal Form:

$- 0.\overline{571428}$