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प्रश्न
Discuss the continuity of the function f, where f is defined by `f(x) = {(-2,"," if x <= -1),(2x, "," if -1 < x <= 1),(2, "," if x > 1):}`
उत्तर
`f(x) = {(-2,"," if x <= -1),(2x, "," if -1 < x <= 1),(2, "," if x > 1):}`
f(x) = -2 for x < 1;
-1 < x < 1, f(x) = 2x and
x > 1, f(x) = 2 is a polynomial function.
So this is a function.
At x = - 1,
`lim_(x -> 1^-) f(x) = lim_(x -> 1^-)` (-2) = -2
`lim_(x -> 1^+) f(x) = lim_(x -> 1^+)` (2x)
= `lim_(h -> 0)` [2 (-1 + h)]
= `lim_(h -> 0)` (-2 + 2h)
= - 2 + 0
= - 2
f(-1) = -2
Hence, f is continuous at x = -1.
At x = 1,
`lim_(x -> 1^-) f(x) = lim_(x -> 1^-)` (2x)
= `lim_(h -> 0)` [2(1 - h)]
= `lim_(h -> 0)` (2 - 2h) = 2 - 2 x 0 = 2
`lim_(x -> 1^+) f(x) = lim_(x -> 1^+)` (2) = 2
f(1) = 2 x 1 = 2
Hence, f is continuous at x = 1.
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