Calculus Examples

$lim_{x \to - 1} \sqrt{u^{2} x^{2} + 2 x u + 1}$

Step 1

Move the limit under the radical sign.

$\sqrt{lim_{x \to - 1} u^{2} x^{2} + 2 x u + 1}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 1$ .

$\sqrt{lim_{x \to - 1} u^{2} x^{2} + lim_{x \to - 1} 2 x u + lim_{x \to - 1} 1}$

Step 3

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

$\sqrt{u^{2} lim_{x \to - 1} x^{2} + lim_{x \to - 1} 2 x u + lim_{x \to - 1} 1}$

Step 4

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

$\sqrt{u^{2} {(lim_{x \to - 1} x)}^{2} + lim_{x \to - 1} 2 x u + lim_{x \to - 1} 1}$

Step 5

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

$\sqrt{u^{2} {(lim_{x \to - 1} x)}^{2} + 2 u lim_{x \to - 1} x + lim_{x \to - 1} 1}$

Step 6

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

$\sqrt{u^{2} {(lim_{x \to - 1} x)}^{2} + 2 u lim_{x \to - 1} x + 1}$

Step 7

Evaluate the limits by plugging in $- 1$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $x$ by plugging in $- 1$ for $x$ .

$\sqrt{u^{2} {(- 1)}^{2} + 2 u lim_{x \to - 1} x + 1}$

Step 7.2

Evaluate the limit of $x$ by plugging in $- 1$ for $x$ .

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

Step 8

Simplify the answer.

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Step 8.1

Rewrite $u^{2} {(- 1)}^{2}$ as ${(u \cdot - 1)}^{2}$ .

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

Step 8.2

Let $u = u \cdot - 1$ . Substitute $u$ for all occurrences of $u \cdot - 1$ .

$\sqrt{u^{2} + 2 u + 1}$

Step 8.3

Factor using the perfect square rule.

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Step 8.3.1

Rewrite $1$ as $1^{2}$ .

$\sqrt{u^{2} + 2 u + 1^{2}}$

Step 8.3.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 u = 2 \cdot u \cdot 1$

Step 8.3.3

Rewrite the polynomial.

$\sqrt{u^{2} + 2 \cdot u \cdot 1 + 1^{2}}$

Step 8.3.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = u$ and $b = 1$ .

$\sqrt{{(u + 1)}^{2}}$

Step 8.4

Replace all occurrences of $u$ with $u \cdot - 1$ .

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

Step 8.5

Simplify each term.

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Step 8.5.1

Move $- 1$ to the left of $u$ .

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

Step 8.5.2

Rewrite $- 1 u$ as $- u$ .

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

Step 8.6

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

$- (- u + 1)$

Step 8.7

Apply the distributive property.

$- - u - 1 \cdot 1$

Step 8.8

Multiply $- - u$ .

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Step 8.8.1

Multiply $- 1$ by $- 1$ .

$1 u - 1 \cdot 1$

Step 8.8.2

Multiply $u$ by $1$ .

$u - 1 \cdot 1$

Step 8.9

Multiply $- 1$ by $1$ .

$u - 1$