Questions

A function is defined as $\mathrm{f}\left(\mathrm{x}\right)=\underset{\mathrm{n}\to \infty }{\mathrm{lim}} \left\{\begin{array}{l}{\mathrm{cos}}^{2\mathrm{n}}\mathrm{x}, \mathrm{if} \mathrm{x}<0\\ \sqrt[\mathrm{n}]{1+{\mathrm{x}}^{\mathrm{n}}}, \mathrm{if} 0\le \mathrm{x}\le 1\\ \frac{1}{1+{\mathrm{x}}^{\mathrm{n}}}, \mathrm{if} \mathrm{x}>1\end{array}\right\$which of the following does not hold good?

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By Expert Faculty of Sri Chaitanya
a
Continuous at x = 0 but discontinuous at x = 1
b
Continuous at x = 1 but discontinuous at x = 0
c
Continuous both at x = 1 and x = 0
d
Discontinuous both at x = 1 and x = 0
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detailed solution

Correct option is A

f0-=limn→∞  limn→0-  cos2xn               = a  value  lesser  than  1∞=0f0+= limn→∞  limn→0+  1+xn1n=1Also, f(0) = 1. So, the function is discontinuous at x = 0.Further,  f1-=1;  f1+=0;  f1=1.So, the function is discontinuous at x = 1.

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$\mathrm{If} \mathrm{f}\left(\mathrm{x}\right)=\frac{{\mathrm{x}}^{2}-\mathrm{bx}+25}{{\mathrm{x}}^{2}-7\mathrm{x}+10} \mathrm{for} \mathrm{x}\ne 5$ is continuous at x = 5, then the value of f(5) is

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