Calculus Examples

$lim_{x \to 0} \sqrt{x^{2} - x^{3}} (1 - \sin^{2} (\frac{2 π}{x}))$

Step 1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$lim_{x \to 0} \sqrt{x^{2} - x^{3}} \cdot lim_{x \to 0} 1 - \sin^{2} (\frac{2 π}{x})$

Step 2

Move the limit under the radical sign.

$\sqrt{lim_{x \to 0} x^{2} - x^{3}} \cdot lim_{x \to 0} 1 - \sin^{2} (\frac{2 π}{x})$

Step 3

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

$\sqrt{lim_{x \to 0} x^{2} - lim_{x \to 0} x^{3}} \cdot lim_{x \to 0} 1 - \sin^{2} (\frac{2 π}{x})$

Step 4

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

$\sqrt{{(lim_{x \to 0} x)}^{2} - lim_{x \to 0} x^{3}} \cdot lim_{x \to 0} 1 - \sin^{2} (\frac{2 π}{x})$

Step 5

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$\sqrt{{(lim_{x \to 0} x)}^{2} - {(lim_{x \to 0} x)}^{3}} \cdot lim_{x \to 0} 1 - \sin^{2} (\frac{2 π}{x})$

Step 6

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

$\sqrt{{(lim_{x \to 0} x)}^{2} - {(lim_{x \to 0} x)}^{3}} \cdot (lim_{x \to 0} 1 - lim_{x \to 0} \sin^{2} (\frac{2 π}{x}))$

Step 7

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$\sqrt{{(lim_{x \to 0} x)}^{2} - {(lim_{x \to 0} x)}^{3}} \cdot (1 - lim_{x \to 0} \sin^{2} (\frac{2 π}{x}))$

Step 8

Move the exponent $2$ from $\sin^{2} (\frac{2 π}{x})$ outside the limit using the Limits Power Rule.

$\sqrt{{(lim_{x \to 0} x)}^{2} - {(lim_{x \to 0} x)}^{3}} \cdot (1 - {(lim_{x \to 0} \sin (\frac{2 π}{x}))}^{2})$

Step 9

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

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Step 9.1

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

$\sqrt{0^{2} - {(lim_{x \to 0} x)}^{3}} \cdot (1 - {(lim_{x \to 0} \sin (\frac{2 π}{x}))}^{2})$

Step 9.2

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

$\sqrt{0^{2} - 0^{3}} \cdot (1 - {(lim_{x \to 0} \sin (\frac{2 π}{x}))}^{2})$

Step 10

Consider the left sided limit.

$lim_{x \to 0^{-}} \sin (\frac{2 π}{x})$

Step 11

Make a table to show the behavior of the function $\sin (\frac{2 π}{x})$ as $x$ approaches $0$ from the left.

$\begin{array}{c} x & \sin (\frac{2 π}{x}) \\ - 0.1 & 0 \\ - 0.01 & - 0.54407169 \\ - 0.001 & 0.8937534 \\ - 0.0001 & - 0.99742913 \\ - 0.00001 & 0.41991556 \\ - 0.000001 & - 0.22195154 \\ - 0.0000001 & 0.97486841 \\ - 0.00000001 & - 0.78016272 \\ 0 & 0.00000197 \\ 0 & 0.00003499 \end{array}$

Step 12

As the $x$ values approach $0$ , the function values approach $0$ . Thus, the limit of $\sin (\frac{2 π}{x})$ as $x$ approaches $0$ from the left is $0$ .

$0$

Step 13

Consider the right sided limit.

$lim_{x \to 0^{+}} \sin (\frac{2 π}{x})$

Step 14

Make a table to show the behavior of the function $\sin (\frac{2 π}{x})$ as $x$ approaches $0$ from the right.

$\begin{array}{c} x & \sin (\frac{2 π}{x}) \\ 0.1 & 0 \\ 0.01 & - 0.95891572 \\ 0.001 & - 0.17658004 \\ 0.0001 & - 0.21424608 \\ 0.00001 & - 0.75115029 \\ 0.000001 & - 0.99796383 \\ 0.0000001 & 0.06293287 \\ 0.00000001 & 0.37855005 \\ 0 & - 0.00000197 \\ 0 & - 0.00003499 \end{array}$

Step 15

As the $x$ values approach $0$ , the function values approach $0$ . Thus, the limit of $\sin (\frac{2 π}{x})$ as $x$ approaches $0$ from the right is $0$ .

$\sqrt{0^{2} - 0^{3}} \cdot (1 - 0^{2})$

Step 16

Simplify the answer.

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Step 16.1

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

$\sqrt{0 - 0^{3}} \cdot (1 - 0^{2})$

Step 16.2

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

$\sqrt{0 - 0} \cdot (1 - 0^{2})$

Step 16.3

Multiply $- 1$ by $0$ .

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

Step 16.4

Add $0$ and $0$ .

$\sqrt{0} \cdot (1 - 0^{2})$

Step 16.5

Rewrite $0$ as $0^{2}$ .

$\sqrt{0^{2}} \cdot (1 - 0^{2})$

Step 16.6

Pull terms out from under the radical, assuming positive real numbers.

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

Step 16.7

Simplify each term.

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Step 16.7.1

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

$0 \cdot (1 - 0)$

Step 16.7.2

Multiply $- 1$ by $0$ .

$0 \cdot (1 + 0)$

Step 16.8

Add $1$ and $0$ .

$0 \cdot 1$

Step 16.9

Multiply $0$ by $1$ .

$0$