SATHEE: Limit Continuity and Differentiability 1 Question 4

Limit Continuity and Differentiability 1 Question 4

5. $lim_{x \to \frac{π}{4}} \frac{\cot^{3} x - \tan x}{\cos x + \frac{π}{4}}$ is

(2019 Main, 12 Jan I)

(a) $4 \sqrt{2}$

(b) 4

(d) $8 \sqrt{2}$

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Answer:

Correct Answer: 5. (c)

Solution:

$lim_{x \to 0^{-}} \frac{x ([x] + | x |) \sin [x]}{| x |} = lim_{x \to 0^{-}} \frac{x ([x] - x) \sin [x]}{- x}$

$\begin{array}{lr} = lim_{x \to 0^{-}} \frac{x (- 1 - x) \sin (- 1)}{- x} (∵ | x | = - x, if x < 0) \\ = lim_{x \to 0^{-}} \frac{- x (x + 1) \sin (- 1)}{- x} = lim_{x \to 0^{-}} (x + 1) \sin (- 1) \\ = (0 + 1) \sin (- 1) (by direct substitution) \\ = - \sin 1 (∵ lim (- θ) = - \sin θ) \end{array}$

Limit Continuity and Differentiability 1 Question 4

5. $lim_{x \to \frac{π}{4}} \frac{\cot^{3} x - \tan x}{\cos x + \frac{π}{4}}$ is

Answer:

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Limit Continuity and Differentiability 1 Question 4

5. limx→π4cot3⁡x−tan⁡xcos⁡x+π4 is

Answer:

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5. $lim_{x \to \frac{π}{4}} \frac{\cot^{3} x - \tan x}{\cos x + \frac{π}{4}}$ is