- #1

songoku

- 2,319

- 331

- Homework Statement
- Please see below

- Relevant Equations
- Not sure

I also tried to calculate the limit directly (not using geometrical argument at all).

$$\lim_{\frac{r_2}{r_1} \rightarrow 1} \left(\frac{r_1+r_2}{{r_2}^{1.5}-{r_1}^{1.5}}\right)$$

$$=\lim_{\frac{r_2}{r_1} \rightarrow 1} \left(\frac{1+\frac{r_2}{r_1}}{\frac{{r_2}^{1.5}}{r_1}-{r_1}^{0.5}}\right)$$

Then got stuck

I also tried rationalization:

$$\lim_{\frac{r_2}{r_1} \rightarrow 1} \left(\frac{r_1+r_2}{{r_2}^{1.5}-{r_1}^{1.5}}\right) \times \frac{{r_2}^{1.5}+{r_1}^{1.5}}{{r_2}^{1.5}+{r_1}^{1.5}}$$

$$=\lim_{\frac{r_2}{r_1} \rightarrow 1} \left(\frac{(r_1+r_2)({r_2}^{1.5}+{r_1}^{1.5})}{{r_2}^{3}-{r_1}^{3}}\right)$$

$$=\lim_{\frac{r_2}{r_1} \rightarrow 1} \left(\frac{(r_1+r_2)({r_2}^{1.5}+{r_1}^{1.5})}{(r_2-r_1)({r_2}^{2}+r_1 r_2+{r_1}^{2}})\right)$$

Then stuck again

Please give me hint, especially how to use geometrical argument.

Thanks