how do you show that lim((e^(h)-1)/h)=1 as h approches 0 given the series e^x= 1+x+(x^2/2!)+(x^3/3!)...
And given F(x)=e^x how do you find the lim((f(x+h)-f(x))/h)as h approches 0
One problem at a time, please.
Using the series form of e^h:
(e^h -1)/h = 1 + h/2! + h^2/3! + ... h^n/(n+1)!
As h--> 0, all of the terms that involve h become zero and only the 1 remains. Therefore that is the limit.
2. That would be the definition of f'(x), which would be e^x = 1.
One can arrive at that result using the series expansion.
lim((f(x+h)-f(x))/h)
x-->0
= (1/h) Lim [1 + (x+h) + (x+h)^2/2! ... - 1 -x -x^2/2! - ...]/h
= Lim 1 + x + (1/3!)[(x+h)^3-x^3] + ...
= 1 + x + x^2/2 + ...
= e^x
To show that lim((e^h - 1)/h) = 1 as h approaches 0, we can make use of the series expansion of e^x. The series expansion of e^x is given by e^x = 1 + x + (x^2/2!) + (x^3/3!) + ...
Let's start by substituting h for x in the series expansion of e^x:
e^h = 1 + h + (h^2/2!) + (h^3/3!) + ...
Now, we'll use this expression to find the limit as h approaches 0 of ((e^h - 1)/h):
lim(h->0) ((e^h - 1)/h) = lim(h->0) (1 + h + (h^2/2!) + (h^3/3!) + ... - 1)/h
Simplifying the expression by canceling out the common terms:
= lim(h->0) (h + (h^2/2!) + (h^3/3!) + ...) / h
Since h is a common factor in the numerator, we can cancel it out:
= lim(h->0) 1 + (h/2!) + (h^2/3!) + ...
Now, as h approaches 0, all terms containing h^2, h^3, and so on become negligible. Therefore, we can ignore them for the purpose of finding the limit:
= lim(h->0) 1 + (h/2!)
And finally, evaluating the limit:
= 1 + 0/2!
= 1
Hence, we have shown that lim((e^h - 1)/h) = 1 as h approaches 0.
Now, let's move on to the second part of the question.
Given f(x) = e^x, we need to find the limit as h approaches 0 of ((f(x + h) - f(x)) / h).
Substituting f(x) = e^x into the expression, we have:
((e^(x+h) - e^x) / h)
Using the exponent rules, we can rewrite e^(x+h) as e^x * e^h:
= ((e^x * e^h - e^x) / h)
Now, we can factor out e^x from the numerator:
= (e^x * (e^h - 1) / h)
Notice that the numerator is exactly the same as the expression we analyzed in the first part of the question. Therefore, using the result we obtained earlier, we know that the limit of ((e^h - 1) / h) as h approaches 0 is 1.
Applying this result to our current expression, the limit becomes:
= (e^x * 1)
= e^x
Therefore, lim((f(x + h) - f(x)) / h) as h approaches 0 is equal to e^x.
Hope this explanation helps!