Nilai limit x mendekati tak hingga (akar(4 x^2+7 x+1-akar(4

\(\frac{11}{4}\)

Pembahasan

Untuk menyelesaikan limit \(\lim_{x\to\infty} (\sqrt{4x^2+7x+1} - \sqrt{4x^2-4x+1})\), kita dapat mengalikan dengan konjugatnya. \(\lim_{x\to\infty} (\sqrt{4x^2+7x+1} - \sqrt{4x^2-4x+1}) \times \frac{\sqrt{4x^2+7x+1} + \sqrt{4x^2-4x+1}}{\sqrt{4x^2+7x+1} + \sqrt{4x^2-4x+1}}\). \(= \lim_{x\to\infty} \frac{(4x^2+7x+1) - (4x^2-4x+1)}{\sqrt{4x^2+7x+1} + \sqrt{4x^2-4x+1}}\). \(= \lim_{x\to\infty} \frac{4x^2+7x+1 - 4x^2+4x-1}{\sqrt{4x^2+7x+1} + \sqrt{4x^2-4x+1}}\). \(= \lim_{x\to\infty} \frac{11x}{\sqrt{4x^2+7x+1} + \sqrt{4x^2-4x+1}}\). Bagi pembilang dan penyebut dengan \(x\) (atau \(\sqrt{x^2}\)): \(= \lim_{x\to\infty} \frac{11}{\sqrt{4+\frac{7}{x}+\frac{1}{x^2}} + \sqrt{4-\frac{4}{x}+\frac{1}{x^2}}}\). Ketika \(x\to\infty\), \(\frac{7}{x}\), \(\frac{1}{x^2}\), \(\frac{4}{x}\) semuanya mendekati 0. \(= \frac{11}{\sqrt{4+0+0} + \sqrt{4-0+0}}\). \(= \frac{11}{\sqrt{4} + \sqrt{4}}\). \(= \frac{11}{2 + 2}\). \(= \frac{11}{4}\).