Question
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\sqrt{1+\text{x}}-\sqrt{1-\text{x}}}{2\text{x}}$
$\lim\limits_{\text{x}\rightarrow0}\frac{\sqrt{1+\text{x}}-\sqrt{1-\text{x}}}{2\text{x}}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{\big(\sqrt{1+\text{x}}-\sqrt{1-\text{x}}\big)}{2\text{x}}\times\frac{\big(\sqrt{1+\text{x}}+\sqrt{1-\text{x}}\big)}{\big(\sqrt{1+\text{x}}+\sqrt{1-\text{x}}\big)}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{(1+\text{x})-{(1-\text{x}})}{2\text{x}\big(\sqrt{1+\text{x}}+\sqrt{1-\text{x}}\big)}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{2\text{x}}{2\text{x}\big(\sqrt{1+\text{x}}+\sqrt{1-\text{x}}\big)}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{1}{\big(\sqrt{1+\text{x}}+\sqrt{1-\text{x}}\big)}$
$=\frac{1}{\sqrt{1}+\sqrt{1}}$
$=\frac{1}{2}$
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| | C1 | | C2 |
| (a) | One book of each subject. | (i) | 3968 |
| (b) | At least one book of each subject. | (ii) | 60 |
| (c) | At least one book of English. | (iii) | 3255 |
| xi | 3 | 8 | 13 | 18 | 23 |
| fi | 7 | 10 | 15 | 10 | 6 |