Question

Find the values of a and b such that the function f defined by
f(x) = `{{:((x - 4)/(|x - 4|) + "a"",",  "if"  x < 4),("a" + "b"",",  "if"  x = 4),((x - 4)/(|x - 4|) + "b"",", "if"  x > 4):}`
is a continuous function at x = 4.

Answer 1

We have, f(x) = `{{:((x - 4)/(|x - 4|) + "a"",",  "if"  x < 4),("a" + "b"",",  "if"  x = 4),((x - 4)/(|x - 4|) + "b"",", "if"  x > 4):}`

At x = 4

L.H.L. = `lim_(x -> 4^-) ((x - 4)/(|x - 4| + "a"))`

= `lim_("h" -> 0) ((4 - "h" - 4)/|4 - "h" - 4| + "a")`

= `lim_("h" -> 0) ((-"h")/"h" + "a")`

= `-1 + "a"`

R.H.L. = `lim_(x -> 4^+) ((x - 4)/|x - 4| + "b")`

= `lim_("h" -> 0) ((4 + "h" - 4)/|4 + "h" - 4| + "b")`

= `lim_("h" -> 0) ("h"/"h" + "b")`

= 1 + b

Also f(4) = a + b  ....(Given)

Since f(x) is continuous at x = 4

–1 + a = 1 + b = a + b

Solving we get, b = –1  and a = 1