can someone please evaluate this limit for me I have been trying for hours.
the original limit was : $$\lim_{x\to \infty} \frac{\sqrt[4]{x+1}-\sqrt[4]{x}}{\sqrt[3]{x+1}-\sqrt[3]{x}}\times{\sqrt[12]{x}}$$
but after some adjustment that I made the limit becomes :
$$\lim_{x\to \infty} \frac{\sqrt[4]{1+\frac{1}{x}}-1}{\sqrt[3]{1+\frac{1}{x}}-1}\times{\frac{\sqrt[4]{x}}{\sqrt[3]{x}}} \times{\sqrt[12]{x}}$$
then $(t=1+\frac{1}{x})$ :
$$\lim_{t\to 1} \frac{\sqrt[4]{t}-1}{\sqrt[3]{t}-1}$$
now I don't know what is the next step
Let $v=\sqrt[12]{t}$. Then your limit is $$\lim_{v\rightarrow 1} \frac{v^3-1}{v^4-1}.$$
Then you can factor out a $v-1$ to get $$\lim_{v\rightarrow 1} \frac{v^2+v+1}{v^3+v^2+v+1} = \frac{3}{4}.$$