- #1

Leo Liu

- 353

- 156

- Homework Statement
- $$\lim_{x\to 0^+}(1-\cos (\sqrt x))^{\sin(x)}$$

- Relevant Equations
- .

I tried taking e^ln but to no avail. Please help! Thanks.

My attempt:

$$\lim_{x\to 0^+}(1-\cos (\sqrt x))^{\sin(x)}$$

$$\lim_{x\to 0^+}e^{\ln (1-\cos\sqrt x)^{\sin x}}$$

$$\lim_{x\to 0^+}e^{{\sin x}\ln (1-\cos\sqrt x)}$$

$$\lim_{x\to 0^+}\exp(\frac{\ln (1-\cos\sqrt x)}{1/\sin x})$$

If I apply Lhospital's rule to this limit, the result will be quite complicated and will remain in intermediate form. I also reached wolfram alpha for help, yet the step by step solution is terse.

My attempt:

$$\lim_{x\to 0^+}(1-\cos (\sqrt x))^{\sin(x)}$$

$$\lim_{x\to 0^+}e^{\ln (1-\cos\sqrt x)^{\sin x}}$$

$$\lim_{x\to 0^+}e^{{\sin x}\ln (1-\cos\sqrt x)}$$

$$\lim_{x\to 0^+}\exp(\frac{\ln (1-\cos\sqrt x)}{1/\sin x})$$

If I apply Lhospital's rule to this limit, the result will be quite complicated and will remain in intermediate form. I also reached wolfram alpha for help, yet the step by step solution is terse.