Calculus Examples

$lim_{h \to 0} \frac{1}{\frac{{(x + h)}^{2}}{h}} - \frac{1}{\frac{x^{2}}{h}}$

Step 1

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

$lim_{h \to 0} \frac{1}{\frac{{(x + h)}^{2}}{h}} - lim_{h \to 0} \frac{1}{\frac{x^{2}}{h}}$

Step 2

Simplify the limit argument.

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Step 2.1

Multiply the numerator by the reciprocal of the denominator.

$lim_{h \to 0} 1 \frac{h}{{(x + h)}^{2}} - lim_{h \to 0} \frac{1}{\frac{x^{2}}{h}}$

Step 2.2

Multiply $\frac{h}{{(x + h)}^{2}}$ by $1$ .

$lim_{h \to 0} \frac{h}{{(x + h)}^{2}} - lim_{h \to 0} \frac{1}{\frac{x^{2}}{h}}$

Step 3

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

$\frac{lim_{h \to 0} h}{lim_{h \to 0} {(x + h)}^{2}} - lim_{h \to 0} \frac{1}{\frac{x^{2}}{h}}$

Step 4

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

$\frac{lim_{h \to 0} h}{{(lim_{h \to 0} x + h)}^{2}} - lim_{h \to 0} \frac{1}{\frac{x^{2}}{h}}$

Step 5

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

$\frac{lim_{h \to 0} h}{{(lim_{h \to 0} x + lim_{h \to 0} h)}^{2}} - lim_{h \to 0} \frac{1}{\frac{x^{2}}{h}}$

Step 6

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

$\frac{lim_{h \to 0} h}{{(x + lim_{h \to 0} h)}^{2}} - lim_{h \to 0} \frac{1}{\frac{x^{2}}{h}}$

Step 7

Simplify the limit argument.

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Step 7.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{lim_{h \to 0} h}{{(x + lim_{h \to 0} h)}^{2}} - lim_{h \to 0} 1 \frac{h}{x^{2}}$

Step 7.2

Multiply $\frac{h}{x^{2}}$ by $1$ .

$\frac{lim_{h \to 0} h}{{(x + lim_{h \to 0} h)}^{2}} - lim_{h \to 0} \frac{h}{x^{2}}$

Step 8

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

$\frac{lim_{h \to 0} h}{{(x + lim_{h \to 0} h)}^{2}} - \frac{1}{x^{2}} lim_{h \to 0} h$

Step 9

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

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Step 9.1

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

$\frac{0}{{(x + lim_{h \to 0} h)}^{2}} - \frac{1}{x^{2}} lim_{h \to 0} h$

Step 9.2

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

$\frac{0}{{(x + 0)}^{2}} - \frac{1}{x^{2}} lim_{h \to 0} h$

Step 9.3

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

$\frac{0}{{(x + 0)}^{2}} - \frac{1}{x^{2}} \cdot 0$

Step 10

Simplify the answer.

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Step 10.1

Combine the opposite terms in $\frac{0}{{(x + 0)}^{2}} - \frac{1}{x^{2}} \cdot 0$ .

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Step 10.1.1

Divide $0$ by ${(x + 0)}^{2}$ .

$0 - \frac{1}{x^{2}} \cdot 0$

Step 10.1.2

Subtract $\frac{1}{x^{2}} \cdot 0$ from $0$ .

$- \frac{1}{x^{2}} \cdot 0$

Step 10.2

Multiply $- \frac{1}{x^{2}} \cdot 0$ .

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Step 10.2.1

Multiply $0$ by $- 1$ .

$0 \frac{1}{x^{2}}$

Step 10.2.2

Multiply $0$ by $\frac{1}{x^{2}}$ .

$0$