![Evaluate the limit: lim (x->0) [sqrt(2x+3) - sqrt(3)] / x](/_next/image?url=https%3A%2F%2Fohijasnjvprzodcmxltt.supabase.co%2Fstorage%2Fv1%2Fobject%2Fpublic%2Fwhatsapp-images%2F1774738575369-625b4246a685c35d.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: Substitute x=0 into the expression to check for an indeterminate form. sqrt(2(0)+3) - sqrt(3)0 = sqrt(3) - sqrt(3)0 = (0)/(0) This is an indeterminate form, so we need to simplify the expression. Step 2: Multiply the numerator and the denominator by the conjugate of the numerator, which is sqrt(2x+3) + sqrt(3). _x 0 sqrt(2x+3) - sqrt(3)x × sqrt(2x+3) + sqrt(3)sqrt(2x+3) + sqrt(3) Step 3: Apply the difference of squares formula (a-b)(a+b) = a^2 - b^2 to the numerator. _x 0 (sqrt(2x+3))^2 - (sqrt(3))^2x(sqrt(2x+3) + sqrt(3)) _x 0 ((2x+3) - 3)/(x(sqrt(2x+3) + 3)) Step 4: Simplify the numerator. _x 0 (2x)/(x(sqrt(2x+3) + 3)) Step 5: Cancel out the common factor x from the numerator and denominator. _x 0 (2)/(sqrt(2x+3) + 3) Step 6: Substitute x=0 into the simplified expression. (2)/(sqrt(2(0)+3) + 3) = (2)/(sqrt(3) + 3) (2)/(2sqrt(3)) Step 7: Simplify the result. (1)/(sqrt(3)) To rationalize the denominator, multiply the numerator and denominator by sqrt(3). (1)/(sqrt(3)) × sqrt(3)sqrt(3) = sqrt(3)3 The final answer is sqrt(3)3.