Question

$\underset{x\to 1}{lim}\frac{1-x^{\frac{1}{3}}}{1-x^{\frac{1}{4}}}=$

• A. $1$
• B. $\frac{4}{3}$
• C. $\frac{3}{4}$
• D. $0$

Answer 1

The correct option is B $\frac{4}{3}$

${lim}_{x\to 1}\frac{1-x^{1/3}}{1-x^{1/4}}\times \frac{x-1}{x-1}$

$={lim}_{x\to 1}\frac{x^{1/3}-1^{1/3}}{x-1}\times \frac{x-1}{x^{1/4}-x^{1/4}}\left[{lim}_{x\to a}\frac{x^n-a^n}{x-a}=n{a}^{n-1}\right]$

$=\frac{1}{3}\cdot {\left(1\right)}^{\frac{1}{3}-1}\times \frac{1}{\frac{1}{4}\cdot {\left(1\right)}^{\frac{1}{4}-1}}=\frac{4}{3}$