Question

for the function f find the value of c if f is continuous at x=0
and evaluate the limit f(x) as x-> infinity

Answer 1

\[3/x^{2} \sin 2x ^{2 }, x <0\]
and
\[(x ^{2} + 2x + c)/1-3x ^{2} , x \ge 0, x \neq 1/\sqrt{3}\]

Answer 2

if the function is continuous at x = 0, then we need two things
1. That the limit exists at x = 0; hence the LH limit = RH limit
2. That the limit equals the value of the function for x = 0

So find the LH and RH limits and determine for what value of c they are equal.

Answer 3

\[\frac{x ^{2} + 2x + c}{1-3x ^{2}} , x \ge 0, x \neq 1/\sqrt{3}\]?

Answer 4

so as x->0 from both sides the functions have the same value?
is that what you mean?

Answer 5

Yes, I mean limit from the RHS (right hand side), or from above, or as x->0+ must equal the limit from the LHS, or from below, or as x->0-

The is a necessary and sufficient condition for the limit to exist at x =0.

As I said above, a function f is continuous at x = 0 if
a) the limit exists at x = 0; and
b) that limit is equal to the value of the function for x = 0.

So you'd better check the limit first. And you'll find btw, that the value of c for which the two limits are equal will also give you the second condition, namely that
\[ \lim_{x \rightarrow 0} f(x) = f(0) \]

Answer 6

correction: "THAT is a necessary and sufficient condition ..."

Answer 7

how do you go about checking a limit ?

Answer 8

You need to go back to your textbook. I'm sure they work in there somewhere an example of a problem like this.

Which is to say, asking me how to evaluate a limit, having being given this problem, is a slightly scary question. Because this question presupposes you know what limits are.

Answer 9

thanks james i believe you are right!