Question
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{3\sin^2\text{x}-2\sin\text{x}^2}{3\text{x}^2}$
$\lim\limits_{\text{x}\rightarrow0}\frac{3\sin^2\text{x}-2\sin\text{x}^2}{3\text{x}^2}$
$=\lim\limits_{\text{x} \rightarrow0}\frac{3\sin^2\text{x}-2\sin\text{x}^2}{3\text{x}^2}-\lim\limits_{\text{x} \rightarrow0}\frac{2\sin\text{x}^2}{3\text{x}^2}$
$=\lim\limits_{\text{x} \rightarrow0}\big(\frac{\sin\text{x}}{\text{x}}\big)^2-\frac{2}{3}\lim\limits_{\text{x} \rightarrow0}\frac{\sin\text{x}^2}{\text{x}^2}$
$=1-\frac{2}{3}\times1$ $\Big[\because\lim\limits_{\theta\rightarrow0}\frac{\sin\theta}{\theta}=1\Big]$
$=1-\frac23$
$=\frac{3-2}{3}$
$=\frac{1}{3}$
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| Column I | Column II | ||
| (a) | $\text{f}-\text{g}$ | (i) | $\Big\{\Big(2, \frac{4}{5}\Big), \Big(8, \frac{-1}{4}\Big), \Big(10, \frac{-3}{13}\Big)\Big\}$ |
| (b) | $\text{f}+\text{g}$ | (ii) | $\{(2, 20), (8, -4), (10, -39)\}$ |
| (c) | $\text{f}\times\text{g}$ | (iii) | $\{(2, 1), (8, -5), (10, -16)\}$ |
| (d) | $\frac{\text{f}}{\text{g}}$ | (iv) | $\{(2, 9), (8, 3), (10, 10)\}$ |
| Marks | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 | 80-90 |
| No. of students | 8 | 10 | 15 | 25 | 20 | 18 | 9 | 5 |
$\text{A}\cup(\text{B}-\text{A})=(\text{A}\cup\text{B})$
$3\text{x}^2+4\text{y}^2-12\text{x}+8\text{y}+4=0$