TS EAMCET · Maths · Continuity and Differentiability

If $f(x)\left\{\begin{array}{cl}\frac{\sqrt{1+k x}-\sqrt{1-k x}}{x}, & \text { for }-1 \leq x < 0 \ 2 x^2+3 x-2, & \text { for } 0 \leq x \leq 1\end{array}\right.$ is continuous at $x=0$, then $k$ is equal to

A -1
B -2
C -3
D -4

Verified Solution

Answer & Solution

Correct Answer

(B) -2

Step-by-step Solution

Detailed explanation

Since, $f(x)$ is continuous at $x=0$ $\therefore \quad \lim _{h \rightarrow 0} f(0-h)=\lim _{h \rightarrow 0} f(0+h)$ $\Rightarrow \lim _{h \rightarrow 0} \frac{\sqrt{1-k h}-\sqrt{1+k h}}{-h}=\lim _{h \rightarrow 0} 2 h^2+3 h-2$…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

Match the items of List-I with those of the items of List-II

	List- $I$		List- $II$
A	Range of $\sec^{- 1} [1 + \cos^{2} x]$ , $[\cdot]$ denote greatest integer function	I	odd function
B	Domain of $f (x)$ where $f (x + \frac{1}{x}) = x^{2} + \frac{1}{x^{2}}$	II	$\{0, \frac{1}{2}\}$
C	$f (x + y) = f (x) + f (y); f (1) = 5$	III	$\{\sec^{- 1} 5, \sec^{- 1} 4\}$
D	$\sin^{- 1} x - \cos^{- 1} x + \sin^{- 1} (1 - x) = 0 \Rightarrow x \in$	IV	$R$
		V	$\{\sec^{- 1} 1, \sec^{- 1} 2\}$

TS EAMCET 2020 Hard

Explore more questions on app

From TS EAMCET

Explore more questions on app

If \(f(x)\left\{\begin{array}{cl}\frac{\sqrt{1+k x}-\sqrt{1-k x}}{x}, & \text { for }-1 \leq x < 0 \ 2 x^2+3 x-2, & \text { for } 0 \leq x \leq 1\end{array}\right.\) is continuous at \(x=0\), then \(k\) is equal to

Answer & Solution

Step-by-step Solution

See the Complete Solution