Question

Evaluate lim x → 0+ x ln(x3). solution the given limit is indeterminate because, as x → 0+, the first factor (x) approaches 0 correct: your answer is correct. while the second factor ln(x3) approaches −∞. writing x = 1/(1/x), we have 1/x → ∞ as x → 0+, so l'hospital's rule gives lim x → 0+ x ln(x3) = lim x → 0+ ln(x3) 1/x = lim x → 0+ 3/x −1/x2 = lim x → 0+ incorrect: your answer is incorrect. = .

Answer 1

`\displaystyle\lim_(x\to0^+)x\ln x^3=\lim_(x\to\infty)\frac{\ln\frac1{x^3}}x=-3\lim_(x\to\infty)\frac{\ln x}x=\frac\infty\infty`

L'Hopital's rule tells us the limit is equal to

`-3\displaystyle\lim_(x\to\infty)\frac{\frac1x}1=0`