SATHEE: Limit Continuity and Differentiability 3 Question 2

Limit Continuity and Differentiability 3 Question 2

2. $lim_{x \to \frac{π}{4}} \frac{\int_{2}^{\sec^{2} x} f (t) d t}{x^{2} - \frac{π^{2}}{16}}$ equals

(a) $\frac{8}{π} f (2)$

(b) $\frac{2}{π} f (2)$

(c) $\frac{2}{π} f \frac{1}{2}$

(d) $4 f (2)$

Show Answer

Answer:

Correct Answer: 2. (1)

Solution:

NOTE All integers are critical point for greatest integer function. Case I When $x \in I$

$f (x) = [x]^{2} - [x^{2}] = x^{2} - x^{2} = 0$

Case II When $x \notin I$ If $0 < x < 1$ , then $[x] = 0$

$and 0 < x^{2} < 1, then [x^{2}] = 0$

Next, if $1 \leq x^{2} < 2 \Rightarrow 1 \leq x < \sqrt{2}$

$\Rightarrow [x] = 1 and [x^{2}] = 1$

Therefore, $f (x) = [x]^{2} - [x^{2}] = 0$ , if $1 \leq x < \sqrt{2}$

Therefore, $f (x) = 0$ , if $0 \leq x < \sqrt{2}$

This shows that $f (x)$ is continuous at $x = 1$ .

Therefore, $f (x)$ is discontinuous in $(- \infty, 0) \cup [\sqrt{2}, \infty)$ on many other points. Therefore, (b) is the answer.

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