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प्रश्न
Examine the continuity of the function f(x) = x3 + 2x2 – 1 at x = 1
उत्तर
We have, f(x) = x3 + 2x2 – 1
For continuity at x = 1
∴ R.H.L. = `lim_(x -> 1^+) "f"(x)`
= `lim_("h" -> 0) "f"(1 + "h")`
= `lim_("h" -> 0) [(1 + "h")^3 + 2(1 + "h")^2 - 1]` = 2
And L.H.L. = `lim_(x -> 1^-) "f"(x)`
= `lim_("h" -> 0) "f"(1 - "h")`
= `lim_("h" -> 0)[(1 - "h")^3 + 2(1 - "h")^2 - 1]` = 2
Also f(1) = 1 + 2 – 1 = 2
Thus `lim_(x -> 1^+) "f"(x) = lim_(x -> 1^-) "f"(x)` = f(1)
Thus f(x) is continuous at x = 1
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