Let f be a function defined on R by f(x)=limn→∞ log⁡(3+x)−x2nsin⁡x1+x2n then

A
$f$ is continuous on R
B
$f$ is continuous on $R ~ {- 1, 1}$
C
$f$ is continuous on $R ~ {0}$
D
none of these

Solution:

If $| x | < 1$ then $x^{2 n} \to 0$ as $n \to \infty$ and if $| x | > 1$

$1 / x^{2 n} \to 0 as n \to \infty$ . So

$f (x) = \{\begin{cases} \log (3 + x) for - 1 < x < 1 \\ - \sin x for x > 1 or x < - 1 \\ \frac{1}{2} [\log (3 + x) - \sin x] for x = 1, - 1 \end{cases}$

$lim_{x \to 1 +} f (x) = - \sin 1, lim_{x \to 1 -} f (x) = \log 4$

$lim_{x \to - 1 +} f (x) = \log 2, lim_{x \to 1 -} f (x) = \sin 1$

so $f$ is continuous on $R ~ {- 1, 1}$

Let $f$ be a function defined on R by $f (x) = lim_{n \to \infty} \frac{\log (3 + x) - x^{2 n} \sin x}{1 + x^{2 n}}$ then

Solution:

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