Studdy AI Logo
Math  /  Calculus

Question32. lim⁡x→02−x−22x\lim _{x \rightarrow 0} \frac{\sqrt{2-x}-\sqrt{2}}{2 x}

Studdy Solution

STEP 1

1. The limit involves a difference of square roots in the numerator.
2. Direct substitution results in an indeterminate form 00 \frac{0}{0} .
3. We will use algebraic manipulation to simplify the expression.

STEP 2

1. Identify the indeterminate form.
2. Rationalize the numerator.
3. Simplify the expression.
4. Evaluate the limit.

STEP 3

First, substitute x=0 x = 0 into the expression to confirm the indeterminate form:
2−0−22⋅0=2−20=00 \frac{\sqrt{2-0}-\sqrt{2}}{2 \cdot 0} = \frac{\sqrt{2} - \sqrt{2}}{0} = \frac{0}{0}
This confirms an indeterminate form.

STEP 4

To resolve the indeterminate form, rationalize the numerator by multiplying the numerator and the denominator by the conjugate of the numerator:
2−x−22x⋅2−x+22−x+2 \frac{\sqrt{2-x}-\sqrt{2}}{2x} \cdot \frac{\sqrt{2-x}+\sqrt{2}}{\sqrt{2-x}+\sqrt{2}}
This gives:
(2−x−2)(2−x+2)2x(2−x+2) \frac{(\sqrt{2-x}-\sqrt{2})(\sqrt{2-x}+\sqrt{2})}{2x(\sqrt{2-x}+\sqrt{2})}

STEP 5

Simplify the numerator using the difference of squares formula:
(2−x)2−(2)2=(2−x)−2=−x (\sqrt{2-x})^2 - (\sqrt{2})^2 = (2-x) - 2 = -x
So the expression becomes:
−x2x(2−x+2) \frac{-x}{2x(\sqrt{2-x}+\sqrt{2})}

STEP 6

Cancel the x x in the numerator and denominator:
−12(2−x+2) \frac{-1}{2(\sqrt{2-x}+\sqrt{2})}

STEP 7

Now, evaluate the limit as x x approaches 0:
lim⁡x→0−12(2−x+2)=−12(2−0+2)=−12(22)=−142 \lim _{x \rightarrow 0} \frac{-1}{2(\sqrt{2-x}+\sqrt{2})} = \frac{-1}{2(\sqrt{2-0}+\sqrt{2})} = \frac{-1}{2(2\sqrt{2})} = \frac{-1}{4\sqrt{2}}
To simplify further, multiply the numerator and the denominator by 2 \sqrt{2} :
−28 \frac{-\sqrt{2}}{8}
The value of the limit is:
−28 \boxed{\frac{-\sqrt{2}}{8}}

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord