Let f (x)= lc 5^ 1 x ; & x<0 [x] ; & x 0, R ., then at x=0

MCQ

Let $f (x)=\left\{\begin{array}{lc}5^{\frac{1}{x}} ; & x<0 \\ \lambda[x] ; & x \geq 0, \lambda \in R \end{array}\right.$, then at $x=0$

✓
f is continuous whatever $\lambda$ may be
B
f is discontinuous
C
f is continuous only if $\lambda=0$
D
none of these

✓

Answer

Correct option: A.

f is continuous whatever $\lambda$ may be

(A)
$\lim _{x \rightarrow 0^{-}} f (x)=\lim _{x \rightarrow 0^{-}} 5^{\frac{1}{x}}=\lim _{ h \rightarrow 0} 5^{-\frac{1}{h}}=0$
$\lim _{x \rightarrow 0^{+}} f (x)=\lim _{x \rightarrow 0^{+}} \lambda[x]=0$, for all $\lambda \in R$
$f(0)=\lambda(0)=0$
$\therefore f$ is continuous at $x=0$, whatever $\lambda$ may be.

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