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
Fresh day Anantaram, let's solve. We need to evaluate the limit: _n sqrt(n)sqrt(n+1) - sqrt(n) The symbol is interpreted as a square root sqrt() due to the context of the problem (limits involving n+1 and n terms, and the options provided). Step 1: Rationalize the denominator. Multiply the numerator and the denominator by the conjugate of the denominator, which is sqrt(n+1) + sqrt(n). _n sqrt(n)sqrt(n+1) - sqrt(n) × sqrt(n+1) + sqrt(n)sqrt(n+1) + sqrt(n) Step 2: Simplify the expression. The denominator uses the difference of squares formula (a-b)(a+b) = a^2 - b^2: (sqrt(n+1) - sqrt(n))(sqrt(n+1) + sqrt(n)) = (sqrt(n+1))^2 - (sqrt(n))^2 = (n+1) - n = 1 The numerator becomes: sqrt(n)(sqrt(n+1) + sqrt(n)) = sqrt(n(n+1)) + sqrt(n^2) = sqrt(n^2+n) + n So the expression simplifies to: _n (sqrt(n^2+n) + n) Step 3: Evaluate the limit. As n , both terms in the expression approach infinity: _n sqrt(n^2+n) = _n n = Therefore, their sum also approaches infinity: _n (sqrt(n^2+n) + n) = + = The limit evaluates to . The final answer is (D) . Send me the next one 📸