In×ln3x ?

The expression "In×ln3x" is not clear as it seems to be a combination of two different mathematical functions.

If you meant to write "ln(3x)", then this represents the natural logarithm of the quantity 3x.

If you meant to write "ln(x) * ln(3x)", then this represents the product of the natural logarithm of x and the natural logarithm of 3x.

Please clarify the expression so that I can provide a more accurate explanation.

Infinite Limits

The infinite limit of In×ln3x as x approaches infinity is ∞. This is because:

- As x approaches infinity, ln3x approaches infinity.
- As ln3x approaches infinity, In×ln3x also approaches infinity.

Therefore, the limit of In×ln3x as x approaches infinity is ∞.

Formal Proof

We can prove this formally using the squeeze theorem:

- Step 1: Find two functions, f(x) and g(x), such that f(x) ≤ In×ln3x ≤ g(x) for all x > some value.
- Step 2: Prove that lim(x->∞) f(x) = lim(x->∞) g(x) = ∞.

For this problem, we can let:

- f(x) = ln3x
- g(x) = x

Then, for all x > 0, we have:

```
ln3x ≤ In×ln3x ≤ x
```

Now, we need to prove that lim(x->∞) ln3x = lim(x->∞) x = ∞. This is easy to show using the definition of the limit:

- For any ε > 0, there exists a number N such that if x > N, then ln3x > ε.
- For any ε > 0, there exists a number N such that if x > N, then x > ε.

Therefore, by the squeeze theorem, lim(x->∞) In×ln3x = ∞.