# Simple limit question. need a little help please

1. Apr 4, 2007

### sutupidmath

i need someone to coment on this:

lim(a^x_n)^x'_n, when n->infinity = (a^lim x_n)^lim x'_n , n-> infinity,

what i am asking here is if we can go from the first to the second??? or if these two expressions are equal??

any help??

2. Apr 4, 2007

### HallsofIvy

Staff Emeritus
Yes, because the exponential function, ax, is continuous.

3. Apr 4, 2007

### sutupidmath

one more thing here. Is there a theorem or a deffinition that supports similar expressions in a more generalized way?? i forgot to mention this also

4. Apr 4, 2007

### HallsofIvy

Staff Emeritus
I'm not sure what you mean. I was referring to the general fact that, from the definition of "continuous", if xn is a sequence of numbers converging to a and f is a function continuous at a, then
$$\lim_{n \rightarrow \infnty} f(x_n)= f(\lim_{n\rightarrow \infty} x_n)= f(a)$$

5. Apr 4, 2007

### sutupidmath

yeah this is what i am asking. But what i want to know is if there is a theorem that states this, what you wrote. Or how do we know that this is so?

6. Apr 5, 2007

### cybercrypt13

Maybe the squeeze theorem?

glenn

7. Apr 6, 2007

### Data

As Halls indicated, it's essentially the definition of continuity. The definition of continuity for a function f of one real variable defined on an interval (a,b) is for any x in (a,b),

$$\lim_{y \rightarrow x} f(y) = f(x)$$

(ie. the limit exists and is equal to f(x))

8. Apr 6, 2007

### sutupidmath

Yeah, i know the definition of continuity, i was just wondering if there is a specific theorem that states this, as i have not encountered one on my calculus book. However, i do understand it now.
Many thanks to all of you.