If f(x)= begincasesx+(1/2), x

Question

If f(x)= begincasesx+(1/2), x<0 2 x+(3/4), x ≥ 0 endcases, then [ displaystyle lim x arrow 0 f(x)]=( where [.] denotes the greatest integer function) (A) (1/2) (B) (3/4) (C) does not exist (D) none of these

Tardigrade Admin · Accepted Answer

displaystyle lim x arrow 0- f(x)= displaystyle lim x arrow 0-(x+(1/2))=(1/2) displaystyle lim x arrow 0+ f(x)= displaystyle lim x arrow 0+(2 x+(3/4))=(3/4) ∴ displaystyle lim x arrow 0 f(x) does not exist.

Q. If $f (x) = {x + \frac{1}{2}, 2 x + \frac{3}{4}, x < 0 x \geq 0$ ,
then $[x \to 0 lim f (x)] = ($ where [.] denotes the greatest integer function)

$\frac{1}{2}$

$\frac{3}{4}$

does not exist

none of these

$x \to 0^{-} lim f (x) = x \to 0^{-} lim (x + \frac{1}{2}) = \frac{1}{2}$
$x \to 0^{+} lim f (x) = x \to 0^{+} lim (2 x + \frac{3}{4}) = \frac{3}{4}$
$∴ x \to 0 lim f (x)$ does not exist.

Q. If f(x)={x+21​,2x+43​,​x<0x≥0​, then [x→0lim​f(x)]=( where [.] denotes the greatest integer function)

21​

43​