![Evaluate the limit: lim (x->0) [sqrt(1+x) - sqrt(1-x)] / x](/_next/image?url=https%3A%2F%2Fohijasnjvprzodcmxltt.supabase.co%2Fstorage%2Fv1%2Fobject%2Fpublic%2Fwhatsapp-images%2F1777910350412-90a57c78e0d7ba8a.png&w=3840&q=75)
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Check for indeterminate form. Substitute x=0 into the expression: sqrt(1+0) - sqrt(1-0)0 = sqrt(1) - sqrt(1)0 = (1-1)/(0) = (0)/(0) This is an indeterminate form, so we need to simplify the expression. Step 2: Multiply by the conjugate of the numerator. Multiply the numerator and the denominator by (sqrt(1+x) + sqrt(1-x)): _x 0 sqrt(1+x) - sqrt(1-x)x × sqrt(1+x) + sqrt(1-x)sqrt(1+x) + sqrt(1-x) Step 3: Simplify the numerator using the difference of squares formula (a-b)(a+b) = a^2 - b^2. Here, a = sqrt(1+x) and b = sqrt(1-x). _x 0 (sqrt(1+x))^2 - (sqrt(1-x))^2x(sqrt(1+x) + sqrt(1-x)) _x 0 ((1+x) - (1-x))/(x(sqrt(1+x) + 1-x)) _x 0 (1+x-1+x)/(x(sqrt(1+x) + 1-x)) _x 0 (2x)/(x(sqrt(1+x) + 1-x)) Step 4: Cancel out the common factor x. Since x 0, x ≠ 0, so we can cancel x from the numerator and denominator: _x 0 (2)/(sqrt(1+x) + 1-x) Step 5: Substitute x=0 into the simplified expression. (2)/(sqrt(1+0) + 1-0) = (2)/(sqrt(1) + 1) = (2)/(1+1) = (2)/(2) = 1 The limit is 1. 3 done, 2 left today. You're making progress.