TS EAMCET · Maths · Continuity and Differentiability
Let \(f\) and \(g\) be real-valued functions. If \(\lim _{x \rightarrow 0} \frac{2 f(x)-g(x)}{[f(x)+7)]^{2 / 3}}=\frac{7}{4}, \lim _{x \rightarrow 0} f(x)=1\) and \(\lim _{x \rightarrow 0} g(x)=\alpha\), then \(h(x)= \begin{cases}\sin (\alpha x), & 0 \leq x \leq \frac{\pi}{10} \ \cos (2 \alpha x), & \frac{\pi}{10} < x \leq \frac{\pi}{5}\end{cases}\)
- A continuous at \(x=\frac{\pi}{10}\) only
- B discontinuous on \(\left[0, \frac{\pi}{5}\right]\)
- C discontinuous at \(x=\frac{\pi}{10}\)
- D continuous on \(\left[0, \frac{\pi}{5}\right]\)
Answer & Solution
Correct Answer
(D) continuous on \(\left[0, \frac{\pi}{5}\right]\)
Step-by-step Solution
Detailed explanation
It is given that \(\lim _{x \rightarrow 0} f(x)=1, \lim _{x \rightarrow 0} g(x)=\alpha\) and \(\lim _{x \rightarrow 0} \frac{2 f(x)-g(x)}{(f(x)+7)^{2 / 3}}=\frac{7}{4}\) \(\Rightarrow \quad \frac{2-\alpha}{(8)^{2 / 3}}=\frac{7}{4} \Rightarrow \alpha=-5\) For…
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