Question

If f(x) = {ln x for 0 < x less than or equal to 2
{x^2 ln2 for 2 < x less than or equal to 4, then the limit as x approaches 2 of f(x) is...

Answer 1

We check the limits as x approaches two from the left and right:

\[\lim_{x \rightarrow 2^-} f(x) = \lim_{x \rightarrow 2^-} \ln x = \ln 2\]
\[\lim_{x \rightarrow 2^+} f(x)=\lim_{x \rightarrow 2^+} x^2 \ln x = 4 \ln 2\]

Since we have two different limits on each side, the limit does not exist.

Answer 2

*A different limit on each side. Is there an edit button?

Answer 3

Let's look at the left limit and the right limit. The left limit is
\[\lim_{x \rightarrow 2} \ln x\]. The answer to this is obviously ln(2).

The right limit is \[\lim_{x \rightarrow 2} \x^2 * ln 2\] = 4*ln(2).

The limit exists only if the left and right limits are the same; since they are not, the limit does not exist.

Answer 4

The 'x' under the 'ln' in my right limit should be a 2 - but the result follows anyway

Answer 5

right, I understand. Thanks to both of you