Find f(a), if f is continuous at x = a where,

f(x) = `(1 + cos(pi x))/(pi(1 - x)^2)`, for x ≠ 1 and at a = 1

#### Solution

f(x) is continuous at x = 1

∴ f(1) = `lim_(x -> 1) "f"(x)`

∴ f(1) = `lim_(x -> 1) (1 + cos pix)/(pi(1 - x)^2`

Put 1 – x = h

∴ x = 1 – h

As x → 1, h → 0

∴ f(1) = `lim_("h" -> 0) (1 + cos[pi(1 - "h")])/(pi"h"^2)`

= `lim_("h" -> 0) (1 + cos(pi - pi"h"))/(pi"h"^2)`

= `lim_("h" -> 0) (1 - cos pi"h")/(pi"h"^2)`

= `lim_("h" -> 0) (1 - cos pi"h")/(pi"h"^2) xx (1 + cos pi"h")/(1 + cos pi"h")`

= `lim_("h" -> 0) (1 - cos^2 pi"h")/(pi"h"^2 (1 + cos pi"h")`

= `1/pi lim_("h" -> 0) (sin^2 pi"h")/("h"^2 (1 + cos pi"h"))`

= `1/pi lim_("h" -> 0) ((sin pi"h")/"h")^2 xx 1/(1 + cos pi"h")`

= `1/pi lim_("h" -> 0) ((sin pi"h")/(pi"h"))^2 xx pi^2 xx 1/(lim_("h" -> 0) (1 + cos pi"h"))`

= `1/pi xx (1)^2 xx pi^2 xx 1/(1 + 1) ...[("As" "h" -> 0"," pi"h" -> 0),("and" lim_(theta -> 0) sintheta/theta = 1)]`

= `pi/2`