Calculus Examples

$lim_{h \to 0} \frac{13 - \frac{π}{h^{2}}}{5 - \frac{6}{h^{2}}}$

Step 1

Combine terms.

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Step 1.1

To write $13$ as a fraction with a common denominator, multiply by $\frac{h^{2}}{h^{2}}$ .

$lim_{h \to 0} \frac{\frac{13 h^{2}}{h^{2}} - \frac{π}{h^{2}}}{5 - \frac{6}{h^{2}}}$

Step 1.2

Combine the numerators over the common denominator.

$lim_{h \to 0} \frac{\frac{13 h^{2} - π}{h^{2}}}{5 - \frac{6}{h^{2}}}$

Step 1.3

To write $5$ as a fraction with a common denominator, multiply by $\frac{h^{2}}{h^{2}}$ .

$lim_{h \to 0} \frac{\frac{13 h^{2} - π}{h^{2}}}{\frac{5 h^{2}}{h^{2}} - \frac{6}{h^{2}}}$

Step 1.4

Combine the numerators over the common denominator.

$lim_{h \to 0} \frac{\frac{13 h^{2} - π}{h^{2}}}{\frac{5 h^{2} - 6}{h^{2}}}$

Step 2

Simplify the limit argument.

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Step 2.1

Multiply the numerator by the reciprocal of the denominator.

$lim_{h \to 0} \frac{13 h^{2} - π}{h^{2}} \cdot \frac{h^{2}}{5 h^{2} - 6}$

Step 2.2

Multiply $\frac{13 h^{2} - π}{h^{2}}$ by $\frac{h^{2}}{5 h^{2} - 6}$ .

$lim_{h \to 0} \frac{(13 h^{2} - π) h^{2}}{h^{2} (5 h^{2} - 6)}$

Step 2.3

Cancel the common factor of $h^{2}$ .

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Step 2.3.1

Cancel the common factor.

$lim_{h \to 0} \frac{(13 h^{2} - π) h^{2}}{h^{2} (5 h^{2} - 6)}$

Step 2.3.2

Rewrite the expression.

$lim_{h \to 0} \frac{13 h^{2} - π}{5 h^{2} - 6}$

Step 3

Split the limit using the Limits Quotient Rule on the limit as $h$ approaches $0$ .

$\frac{lim_{h \to 0} 13 h^{2} - π}{lim_{h \to 0} 5 h^{2} - 6}$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$\frac{lim_{h \to 0} 13 h^{2} - lim_{h \to 0} π}{lim_{h \to 0} 5 h^{2} - 6}$

Step 5

Move the term $13$ outside of the limit because it is constant with respect to $h$ .

$\frac{13 lim_{h \to 0} h^{2} - lim_{h \to 0} π}{lim_{h \to 0} 5 h^{2} - 6}$

Step 6

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

$\frac{13 {(lim_{h \to 0} h)}^{2} - lim_{h \to 0} π}{lim_{h \to 0} 5 h^{2} - 6}$

Step 7

Evaluate the limit of $π$ which is constant as $h$ approaches $0$ .

$\frac{13 {(lim_{h \to 0} h)}^{2} - π}{lim_{h \to 0} 5 h^{2} - 6}$

Step 8

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$\frac{13 {(lim_{h \to 0} h)}^{2} - π}{lim_{h \to 0} 5 h^{2} - lim_{h \to 0} 6}$

Step 9

Move the term $5$ outside of the limit because it is constant with respect to $h$ .

$\frac{13 {(lim_{h \to 0} h)}^{2} - π}{5 lim_{h \to 0} h^{2} - lim_{h \to 0} 6}$

Step 10

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

$\frac{13 {(lim_{h \to 0} h)}^{2} - π}{5 {(lim_{h \to 0} h)}^{2} - lim_{h \to 0} 6}$

Step 11

Evaluate the limit of $6$ which is constant as $h$ approaches $0$ .

$\frac{13 {(lim_{h \to 0} h)}^{2} - π}{5 {(lim_{h \to 0} h)}^{2} - 1 \cdot 6}$

Step 12

Evaluate the limits by plugging in $0$ for all occurrences of $h$ .

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Step 12.1

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$\frac{13 \cdot 0^{2} - π}{5 {(lim_{h \to 0} h)}^{2} - 1 \cdot 6}$

Step 12.2

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$\frac{13 \cdot 0^{2} - π}{5 \cdot 0^{2} - 1 \cdot 6}$

Step 13

Simplify the answer.

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Step 13.1

Simplify the numerator.

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Step 13.1.1

Raising $0$ to any positive power yields $0$ .

$\frac{13 \cdot 0 - π}{5 \cdot 0^{2} - 1 \cdot 6}$

Step 13.1.2

Multiply $13$ by $0$ .

$\frac{0 - π}{5 \cdot 0^{2} - 1 \cdot 6}$

Step 13.1.3

Subtract $π$ from $0$ .

$\frac{- π}{5 \cdot 0^{2} - 1 \cdot 6}$

Step 13.2

Simplify the denominator.

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Step 13.2.1

Raising $0$ to any positive power yields $0$ .

$\frac{- π}{5 \cdot 0 - 1 \cdot 6}$

Step 13.2.2

Multiply $5$ by $0$ .

$\frac{- π}{0 - 1 \cdot 6}$

Step 13.2.3

Multiply $- 1$ by $6$ .

$\frac{- π}{0 - 6}$

Step 13.2.4

Subtract $6$ from $0$ .

$\frac{- π}{- 6}$

Step 13.3

Dividing two negative values results in a positive value.

$\frac{π}{6}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{π}{6}$

Decimal Form:

$0.52359877 \dots$