(a) यहाँ `f(x)=(|sin x|)/(x),`
(i) `f(0)=1`
(ii) R.H.L. `=underset(x rarr 0^(+))(lim)f(x)=underset(h rarr 0)(lim)f(0+h)`
`=underset(h rarr 0)(lim)(|sin (0 + h)|)/((0+h)),” “[because x = 0 + h ne 0]`
`=underset(h rarr 0)(lim)(|sin h|)/(h)`
`=underset(h rarr 0)(lim)(sin h)/(h)`
`= 1″ “[because underset(theta rarr 0)(lim)(sin theta)/(theta)=1]`
(iii) L.H.L. `=underset(x rarr 0^(-))(lim)f(x)=underset(h rarr 0)(lim)f(0-h)`
`=underset(h rarr 0)(lim)(|sin (0-h)|)/((0-h))`
`=underset(h rarr 0)(lim)(|sin (-h)|)/(-h)`
`=underset(h rarr 0)(lim)(|-sin h|)/(-h),” “[because sin (-theta)=sin theta]`
`=underset(h rarr 0)(lim)(sin h)/(-h)`
`=-1″ “[because underset(theta rarr 0)(lim)(sin theta)d/(theta)=1]`
`therefore” “underset(x rarr 0^(-))(lim)f(x) ne underset(x rarr 0^(+))(lim)f(x)=f(0)`
अत: `f(x), x = 0` पर असंतत है ।
(b) यहाँ `f(x)=x “sin”(1)/(x)`
(i) `f(0)=0`
(ii) R.H.L. `=underset(x rarr 0^(+))(lim)f(x)=underset(h rarr 0)(lim)f(0+h)`
`=underset(h rarr 0)(lim)(0+h)”sin”(1)/((0+h))`
`=underset(h rarr 0)(lim)h sin((1)/(h))`
`= 0 xx` परिमिति राशि, [`because “sin”(1)/(h)` का मान -1 और +1 के मध्य स्थित होता है]
= 0
(iii) L.H.L. `=underset(x rarr 0^(-))(lim)f(x)=underset(h rarr 0)(lim)f(0-h)`
`=underset(h rarr 0)(lim)(0-h)sin((1)/(0-h))`
`=underset(h rarr 0)(lim)(-h)sin(-(1)/(h))`
`=underset(h rarr 0)(lim)(-h)(-sin((1)/(h)))” “[because sin(-theta)=-sin theta]`
`=underset(h rarr 0)(lim)hsin((1)/(h))`
`=0 xx` अपरिमित राशि,
[`because “sin”(1)/(h)` का मान -1 और +1 के मध्य स्थित होता है]
= 0
`therefore” “underset(x rarr 0^(+))(lim)f(x)=underset(x rarr 0^(-))(lim)f(x)=f(0)`
अत: `f(x), x = 0` पर संतत है ।

(a) यहाँ `f(x)={{:((1-cos x)/(x^(2))”,”,x ne 0),(” “(1)/(2)”,”,x = 0):}`
(i) `f(0) = (1)/(2)`
(ii) R.H.L. `=underset(x rarr 0^(+))(lim)f(x)=underset(h rarr 0)(lim)f(0+h)`
`=underset(h rarr 0)(lim)(1-cos(0+h))/((0+h)^(2))`
`=underset(h rarr 0)(lim)(1-cos h)/(h^(2))`
`=underset(h rarr 0)(lim)(1-(1-2 “sin”^(2) (h)/(2)))/(h^(2))” “[because cos x = 1 -2 “sin”^(2)(x)/(2)]`
`=underset(h rarr 0)(lim)(2″sin”^(2)(h)/(2))/(h^(2))`
`=2 underset(h rarr 0)(lim)((“sin”(h)/(2))/((h)/(2)))^(2)xx (1)/(4)`
`=2 xx (1)^(2)xx(1)/(4),” “[because underset(theta rarr 0)(lim)(sin theta)/(theta)=1]`
`=(1)/(2)`
(iii) L.H.L. `=underset(x rarr 0^(-))(lim)f(x)=underset(h rarr 0)(lim)f(0-h)`
`=underset(h rarr 0)(lim)(1-cos(0-h))/((0-h)^(2))`
`=underset(h rarr 0)(lim)(1-cos h)/(h^(2))” “[because cos (-theta)=cos theta]`
`=underset(h rarr 0)(lim)(2″sin”^(2)(h)/(2))/(h^(2))`
`=2 underset(h rarr 0)(lim)((“sin”(h)/(2))/((h)/(2)))^(2)xx(1)/(4)`
`=(1)/(2)`
`therefore” “underset(x rarr 0^(+))(lim)f(x)=underset(x rarr 0^(-))(lim)f(x)=f(0)`
अत: `f(x), x = 0` पर संतत है ।
(b) यहाँ `f(x)={{:((sin ax)/(sin bx)”,”,x ne 0),((a)/(b)”,”,x = 0):}`
(i) `f(0) = (a)/(b)`
(ii) R.H.L. `=underset(x rarr 0^(+))(lim)f(x)=underset(h rarr 0)(lim)f(0+h)`
`=underset(h rarr 0)(lim)(sin a(0+h))/(sin b(0+h))`
`=underset(h rarr 0)(lim)(sin ah)/(sin bh)`
`=underset(h rarr 0)(lim)[(sin ah)/(ah)xxahxx(bh)/(sinbh)xx(1)/(bh)]`
`=(a)/(b)[underset(h rarr 0)(lim)(sin ah)/(ah)xx underset(h rarr 0)(lim)(1)/(((sin bh)/(bh)))]`
`=(a)/(b)xx 1 xx 1,” “[because h rarr 0 rArr ah rarr “0 और bh” rarr 0]`
`=(a)/(b)`
(iii) L.H.L. `=underset(x rarr 0^(-))(lim)f(x)=underset(h rarr 0)(lim)f(0-h)`
`=underset(h rarr 0)(lim)(sin a(0-h))/(sin b(0-h))`
`=underset(h rarr 0)(lim)(sin(-ah))/(sin(-bh))`
`= underset(h rarr 0)(lim)(sin ah)/(sin bh)`
`=(a)/(b)`
`therefore” “underset(x rarr 0^(+))(lim)f(x)=underset(x rarr 0^(-))(lim)f(x)=f(0)=(a)/(b)`
अत: `f(x), x = 0` पर संतत है ।