Question
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow1}\frac{{\text{x}-1}}{\sqrt{\text{x}^2+3}-2}$
$\lim\limits_{\text{x}\rightarrow1}\frac{{\text{x}-1}}{\sqrt{\text{x}^2+3}-2}$
$=\lim\limits_{\text{x}\rightarrow1}\frac{(\text{x}-1)\times\big(\sqrt{\text{x}^2+3}+2\big)}{\big(\sqrt{\text{x}^2+3}-2\big)\big(\sqrt{\text{x}^2+3}+2\big)}$
$=\lim\limits_{\text{x}\rightarrow1}\frac{(\text{x}-1)\big(\sqrt{\text{x}^2+3}+2\big)}{\big({\text{x}^2+3}-4\big)}$
$=\lim\limits_{\text{x}\rightarrow1}\frac{(\text{x}-1)\big(\sqrt{\text{x}^2+3}+2\big)}{\big({\text{x}^2-1}\big)}$
$=\lim\limits_{\text{x}\rightarrow1}\frac{\sqrt{\text{x}^2+3}+2}{{\text{x}}+1}$
Putting the value x = 1
$\Rightarrow\frac{\sqrt{1+3}+2}{1+1}$
$=\frac{2+2}{2}$
$=\frac{4}{2}=2$
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
| Diameters | 33-36 | 37-40 | 41-44 | 45-48 | 49-52 |
| No. of circles | 15 | 17 | 21 | 22 | 25 |
Calculate the standard deviation and mean diameter of the circles.
[Hint First make the data continuous by making the classes as 32.5-36.5, 36.5-40.5, 40.5-44.5, 44.5 - 48.5, 48.5 - 52.5 and then proceed.]
| (i) | $((\text{A}'\cup\text{B}')-\text{A})'$ | (a) | $\text{A} - \text{B}$ |
| (ii) | $[\text{B}'\cup(\text{B}'-\text{A})]'$ | (b) | $\text{A}$ |
| (iii) | $(\text{A} - \text{B}) - (\text{B} - \text{C})$ | (c) | $\text{B}$ |
| (iv) | $(\text{A}-\text{B})\cap(\text{C}-\text{B})$ | (d) | $(\text{A}\times\text{B})\cap(\text{A}\times\text{C})$ |
| (v) | $\text{A}\times(\text{B}\cap\text{C})$ | (e) | $(\text{A}\times\text{B})\cup(\text{A}\times\text{C})$ |
| (vi) | $\text{A}\times(\text{B}\cup\text{C})$ | (f) | $(\text{A}\cap\text{C})-\text{B}$ |