Question

Evaluate the limit


`\lim \limits_(x\to 0) (\ln \left(1-\frac x4 \right) - (1-x)^(\tfrac 14) +1)/(x^2)`

Answer 1

1/16

Explanation:

Place x=0 and you can see
`(0)/(0)` indefinitness. So you can apply the l'hosptial rule. Its basic you should

`\lim_(x \to \ 0) (g(x))/(f(x)) = \\`
`lim_(x \to \ 0) (g'(x))/(f'(x))` so apply the derivative

`\frac{(1)/(x-4) +\frac{1}{\sqrt[4]{(1-x})^3 } }{2x}` and replace the x=0 and you'll see same answer
`(0)/(0)` re-apply the l'hospital rule and answer

`\frac{(-1)/((x-4)^2) + \frac{3}{16\sqrt[4]{(1-x)^7} } }{2}` and replace the 0 you can see the answer
`(1)/(16)`