![Evaluate the limit: lim (x->0) [sqrt(1+x2) - 1] / x2](/_next/image?url=https%3A%2F%2Fohijasnjvprzodcmxltt.supabase.co%2Fstorage%2Fv1%2Fobject%2Fpublic%2Fwhatsapp-images%2F1777663844016-82570aacc19fc885.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
Hey, good to see you again. Step 1: Check for indeterminate form by substituting x=0. _x 0 sqrt(1+x^2) - 1x^2 = sqrt(1+0^2) - 10^2 = sqrt(1) - 10 = (1-1)/(0) = (0)/(0) Since this is an indeterminate form, we need to simplify the expression. Step 2: Multiply the numerator and denominator by the conjugate of the numerator. The conjugate of sqrt(1+x^2) - 1 is sqrt(1+x^2) + 1. _x 0 sqrt(1+x^2) - 1x^2 × sqrt(1+x^2) + 1sqrt(1+x^2) + 1 Step 3: Simplify the numerator using the difference of squares formula, (a-b)(a+b) = a^2 - b^2. Here, a = sqrt(1+x^2) and b = 1. _x 0 (sqrt(1+x^2))^2 - 1^2x^2(sqrt(1+x^2) + 1) = _x 0 ((1+x^2) - 1)/(x^2(sqrt(1+x^2) + 1)) Step 4: Further simplify the numerator. _x 0 (x^2)/(x^2(sqrt(1+x^2) + 1)) Step 5: Cancel out the common term x^2 from the numerator and denominator (since x 0, x ≠ 0). _x 0 (1)/(sqrt(1+x^2) + 1) Step 6: Substitute x=0 into the simplified expression to evaluate the limit. (1)/(sqrt(1+0^2) + 1) = (1)/(sqrt(1) + 1) = (1)/(1 + 1) = (1)/(2) The value of the limit is (1)/(2). The final answer is (1)/(2). Send me the next one 📸