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: Identify the highest power of x in the denominator and the effective highest power in the numerator. The highest power in the denominator is x^3. In the numerator, we have sqrt(1 + 4x^6). As x , the 1 becomes negligible compared to 4x^6. So, the dominant term is sqrt(4x^6) = sqrt(4)sqrt(x^6) = 2|x^3|. Since x , x is positive, so |x^3| = x^3. Thus, the effective highest power in the numerator is x^3. Step 2: Divide every term in the numerator and the denominator by x^3. For terms inside the square root in the numerator, dividing by x^3 is equivalent to dividing by sqrt(x^6) (since x , x^3 = sqrt(x^6)). _x sqrt(1 + 4x^6)x^3(2 - x^3)/(x^3) Step 3: Simplify the numerator and the denominator. Numerator: sqrt(1 + 4x^6)x^3 = sqrt(1 + 4x^6)sqrt(x^6) = sqrt((1 + 4x^6)/(x^6)) = sqrt((1)/(x^6) + (4x^6)/(x^6)) = sqrt((1)/(x^6) + 4) Denominator: (2 - x^3)/(x^3) = (2)/(x^3) - (x^3)/(x^3) = (2)/(x^3) - 1 Step 4: Substitute the simplified expressions back into the limit. _x (sqrt(1)/(x^6) + 4)(2)/(x^3) - 1 Step 5: Evaluate the limit by applying the property that _x (c)/(x^n) = 0 for any constant c and positive integer n. As x : (1)/(x^6) 0 (2)/(x^3) 0 So the limit becomes: sqrt(0 + 4)0 - 1 = sqrt(4)-1 = (2)/(-1) = -2 The final answer is -2. 3 done, 2 left today. You're making progress.