How can we test the continuity of the function below :--

Question

maheshmeghwal9 · Answer

$$f(x)=(e^x-1)/x$$ for $$x \in R$$ but$$x \neq0$$ .

radar · Answer

I would say that by inspection, all real numbers except zero.  I may be wrong as expontential functions is not my cup of tea.

maheshmeghwal9 · Answer

ya ok.np:)

anonymous · Answer

The big places to check are sign changes of the different parts of the function.

e^x is always positive, and e^0 =1, so the top goes from negative to positive at x=0

The bottom also changes sign at x=0

So really, 0 is the only place to check continuity Since they exclude that point, I'd say you're doneso.

maheshmeghwal9 · Answer

k!

maheshmeghwal9 · Answer

can u prove it with the help of limit with steps!please

anonymous · Answer

And if you do have places to check, the thing to check is the left limit and right limit at those places.

maheshmeghwal9 · Answer

but how?

alfie · Answer

Notice that this function is always positive. And that your domain is x != 0.

So what you want to do is the limit for x-> 0 , check what it is. You will find out it's 1 (pick your favourite method to solve this).

So what do you find out?

maheshmeghwal9 · Answer

I found 1.but how can we say that it is  continuous function?

alfie · Answer

You have to recall both definition of continuity AND discontinuity. This might help you out: http://www.math.brown.edu/UTRA/discontinuities.html

maheshmeghwal9 · Answer

k! is there no meaning of left hand limit & right hand limit?

radar · Answer

Good link alfie.

alfie · Answer

basically you have to imagine in your head a point, you can either reach it from the left, either from the right, that's what it's really all about. You want to check what happens when "You are going near to that point" from the right / left.

maheshmeghwal9 · Answer

how will we test it by using standard limit--i.e.;$$(a^x-1)/x=\log{a} $$

maheshmeghwal9 · Answer

would anybody please help me?

kropot72 · Answer

For functions of one variable the function y = f(x) is continuous at point x = a if the following 3 conditions are satisfied:

1) f(a) is defined
2)The following limit exists
$$\lim_{x \rightarrow a}f(x)$$
3)$$\lim_{x \rightarrow a}f(x)=f(a)$$

maheshmeghwal9 · Answer

From the above standard limit condition, we get f(x)=$$(e^x-1)/x=\log{e} $$ now , How can I say that $$f(x)=\log{e} $$ is continuous for $$x>0$$

kropot72 · Answer

What was your limit?

maheshmeghwal9 · Answer

$$\lim_{x \rightarrow 0} (e^x-1)/x=\log e$$

kropot72 · Answer

$$x \neq0$$according to the question. Therefore it is not a valid limit.