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प्रश्न
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.
उत्तर
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
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