TS EAMCET · Maths · Continuity and Differentiability
At \(x=0, f(x)=\left\{\begin{array}{l}\frac{x}{|x|+2 x^2}, x \neq 0 \ k, \quad x=0\end{array}\right.\) is
- A Continuous only when \(k=0\)
- B Discontinuous only when \(k=0\)
- C Continuous for all values of \(k\)
- D Discontinuous for all real values of \(k\)
Answer & Solution
Correct Answer
(D) Discontinuous for all real values of \(k\)
Step-by-step Solution
Detailed explanation
Given that, \(f(x)=\left\{\begin{array}{cc}\frac{x}{|x|+2 x^2}, & x \neq 0 \\ k, & x=0\end{array}\right.\) \(\mathrm{LHL}=\lim _{x \rightarrow 0^{-}} f(x)=\lim _{h \rightarrow 0} f(0-h)\) \(=\lim _{h \rightarrow 0} \frac{-h}{|-h|+2(-h)^2}=\frac{-h}{h+2 h^2}=\frac{-h}{h(1+2 h)}\)…
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