Calculus Examples

$lim_{h \to (\frac{3}{7})} \frac{7 h - 3}{2 h}$

Step 1

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

$\frac{1}{2} lim_{h \to (\frac{3}{7})} \frac{7 h - 3}{h}$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $h$ approaches $(\frac{3}{7})$ .

$\frac{1}{2} \cdot \frac{lim_{h \to (\frac{3}{7})} 7 h - 3}{lim_{h \to (\frac{3}{7})} h}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $(\frac{3}{7})$ .

$\frac{1}{2} \cdot \frac{lim_{h \to (\frac{3}{7})} 7 h - lim_{h \to (\frac{3}{7})} 3}{lim_{h \to (\frac{3}{7})} h}$

Step 4

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

$\frac{1}{2} \cdot \frac{7 lim_{h \to (\frac{3}{7})} h - lim_{h \to (\frac{3}{7})} 3}{lim_{h \to (\frac{3}{7})} h}$

Step 5

Evaluate the limit of $3$ which is constant as $h$ approaches $(\frac{3}{7})$ .

$\frac{1}{2} \cdot \frac{7 lim_{h \to (\frac{3}{7})} h - 1 \cdot 3}{lim_{h \to (\frac{3}{7})} h}$

Step 6

Evaluate the limits by plugging in $(\frac{3}{7})$ for all occurrences of $h$ .

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Step 6.1

Evaluate the limit of $h$ by plugging in $(\frac{3}{7})$ for $h$ .

$\frac{1}{2} \cdot \frac{7 (\frac{3}{7}) - 1 \cdot 3}{lim_{h \to (\frac{3}{7})} h}$

Step 6.2

Evaluate the limit of $h$ by plugging in $(\frac{3}{7})$ for $h$ .

$\frac{1}{2} \cdot \frac{7 (\frac{3}{7}) - 1 \cdot 3}{(\frac{3}{7})}$

Step 7

Simplify the answer.

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Step 7.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{2} ((7 (\frac{3}{7}) - 1 \cdot 3) \frac{7}{3})$

Step 7.2

Simplify each term.

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Step 7.2.1

Cancel the common factor of $7$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{1}{2} ((7 (\frac{3}{7}) - 1 \cdot 3) \frac{7}{3})$

Step 7.2.1.2

Rewrite the expression.

$\frac{1}{2} ((3 - 1 \cdot 3) \frac{7}{3})$

Step 7.2.2

Multiply $- 1$ by $3$ .

$\frac{1}{2} ((3 - 3) \frac{7}{3})$

Step 7.3

Subtract $3$ from $3$ .

$\frac{1}{2} (0 (\frac{7}{3}))$

Step 7.4

Cancel the common factor of $2$ .

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Step 7.4.1

Factor $2$ out of $0 (\frac{7}{3})$ .

$\frac{1}{2} (2 (0 (\frac{7}{3})))$

Step 7.4.2

Cancel the common factor.

$\frac{1}{2} (2 (0 (\frac{7}{3})))$

Step 7.4.3

Rewrite the expression.

$0 (\frac{7}{3})$

Step 7.5

Multiply $0$ by $\frac{7}{3}$ .

$0$