Question

lim(x→1)⁡ (∛x-√x)/(∜x-1)

Answer 1

Since plugging in 1 gives us the 0/0 case, we can use L'Hospital's rule to help us solve this limit that says\[\lim_{x \rightarrow a} f(x)\div g(x)\]Where:\[f(a)\div g(a) = 0\div 0\]Then\[\lim_{x \rightarrow a}f(x)\div g(x) = \lim_{x \rightarrow a}f^{\prime}(x)\div g^{\prime}(x) = f^{\prime}(a)\div g^{\prime}(a)\]

Answer 2

To follow up, the answer is\[\lim_{x \rightarrow 1}(\sqrt[3]{x}-\sqrt{x}) \div (\sqrt[4]{x} -1) = -2 \div 3\]But make sure you can arrive at this answer before moving onto the next question. If you need help with finding the derivatives, I'll be glad to help.