Question

If f:R-->R is continuous at x=c and lim x_n (n-->inf) = c, show lim f(x_n) (n-->inf) = f(c)

Answer 1

are you using epsilon-delta, epsilon-N definitions of limits here?

Answer 2

yes, sir.

Answer 3

So you want to show that for all epsilon e > 0 there exists an N such that
\[ n > N \ \implies \ | f(x_n) - f(c) | < \epsilon \]

Answer 4

yes, sir

Answer 5

Now, given the two things you know, write down the formal statements of them. Then see how you have to chain them together to prove the result I just wrote down.

Answer 6

f is cont. @ x=c means lim f(x) (x-->c) = f(c). Now, since lim x_n (n-->inf) = c, therefore, lim f(x_n) (n-->inf) = f(c). Am I right, sir?

Answer 7

Yes, but I assume you need to prove it; what you wrote down is not a proof.