Calculus Examples

$lim_{x \to 0} \sqrt{- 2 x^{3} - 5 x^{2} + 2 x + 2}$

Step 1

Move the limit under the radical sign.

$\sqrt{lim_{x \to 0} - 2 x^{3} - 5 x^{2} + 2 x + 2}$

Step 2

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

$\sqrt{- lim_{x \to 0} 2 x^{3} - lim_{x \to 0} 5 x^{2} + lim_{x \to 0} 2 x + lim_{x \to 0} 2}$

Step 3

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

$\sqrt{- 2 lim_{x \to 0} x^{3} - lim_{x \to 0} 5 x^{2} + lim_{x \to 0} 2 x + lim_{x \to 0} 2}$

Step 4

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

$\sqrt{- 2 {(lim_{x \to 0} x)}^{3} - lim_{x \to 0} 5 x^{2} + lim_{x \to 0} 2 x + lim_{x \to 0} 2}$

Step 5

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

$\sqrt{- 2 {(lim_{x \to 0} x)}^{3} - 5 lim_{x \to 0} x^{2} + lim_{x \to 0} 2 x + lim_{x \to 0} 2}$

Step 6

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

$\sqrt{- 2 {(lim_{x \to 0} x)}^{3} - 5 {(lim_{x \to 0} x)}^{2} + lim_{x \to 0} 2 x + lim_{x \to 0} 2}$

Step 7

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

$\sqrt{- 2 {(lim_{x \to 0} x)}^{3} - 5 {(lim_{x \to 0} x)}^{2} + 2 lim_{x \to 0} x + lim_{x \to 0} 2}$

Step 8

Evaluate the limit of $2$ which is constant as $x$ approaches $0$ .

$\sqrt{- 2 {(lim_{x \to 0} x)}^{3} - 5 {(lim_{x \to 0} x)}^{2} + 2 lim_{x \to 0} x + 2}$

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 $x$ .

$\sqrt{- 2 \cdot 0^{3} - 5 {(lim_{x \to 0} x)}^{2} + 2 lim_{x \to 0} x + 2}$

Step 9.2

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

$\sqrt{- 2 \cdot 0^{3} - 5 \cdot 0^{2} + 2 lim_{x \to 0} x + 2}$

Step 9.3

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

$\sqrt{- 2 \cdot 0^{3} - 5 \cdot 0^{2} + 2 \cdot 0 + 2}$

Step 10

Simplify the answer.

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Step 10.1

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

$\sqrt{- 2 \cdot 0 - 5 \cdot 0^{2} + 2 \cdot 0 + 2}$

Step 10.2

Multiply $- 2$ by $0$ .

$\sqrt{0 - 5 \cdot 0^{2} + 2 \cdot 0 + 2}$

Step 10.3

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

$\sqrt{0 - 5 \cdot 0 + 2 \cdot 0 + 2}$

Step 10.4

Multiply $- 5$ by $0$ .

$\sqrt{0 + 0 + 2 \cdot 0 + 2}$

Step 10.5

Multiply $2$ by $0$ .

$\sqrt{0 + 0 + 0 + 2}$

Step 10.6

Add $0$ and $0$ .

$\sqrt{0 + 0 + 2}$

Step 10.7

Add $0$ and $0$ .

$\sqrt{0 + 2}$

Step 10.8

Add $0$ and $2$ .

$\sqrt{2}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\sqrt{2}$

Decimal Form:

$1.41421356 \dots$