Question

Examine the continuity of the function f(x) = x³ + 2x² – 1 at x = 1

Answer 1

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