Maharashtra State BoardHSC Science (Electronics) 11th
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Find f(a), if f is continuous at x = a where, f(x) = 1+cos(πx)π(1-x)2, for x ≠ 1 and at a = 1 - Mathematics and Statistics

Sum

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

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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`

  Is there an error in this question or solution?
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APPEARS IN

Balbharati Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board
Chapter 8 Continuity
Miscellaneous Exercise 8 | Q VII. (1) | Page 178
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