1. Use the Taylor series to calculate the limit.
Problem: limit as x approaches 0 is equal to (1-cos(x))/(1+x-e^x).
I did the problem out but I need help in seeing if its correct.
limit as x approaches 0 is equal to (1-cos(x))/(1+x-e^x)= (1-(1-x^2/2+x^4/4!-...))/(1+x-(1+x+x^2/2+x^3/3!+x^4/4!+...)=(x^2-x^4/4!+...)/(-x^2/2-x^3/3!-x^4/4!-...)=-11/17
It is wrong. You also need to use the "Big O" notation. One denotes by
O(x^n) a term that for x to zero behaves as a constant times x^n. More rigorously, we say that some term T(x) is O(x^n) if and only if there exists a p and C such that
|T(x)| <= C |x^n| for all |x| < p
THen we can proceed as follows. We have:
1-cos(x) = 1/2 x^2 + O(x^4)
1 +x-exp(x) = -1/2 x^2 + O(x^3)
[1-cos(x)]/[1+x-exp(x)] =
-[1+O(x^2)]/[1+O(x)] =
-[1+O(x^2)][1+O(x)] =
-1 + O(x)
Limit of x to zero is then -1, because the O(x) tends to zero as it is less than x for x sufficiently close to zero.
To use the Taylor series to calculate the limit as x approaches 0 for the given problem, you have made a good start. However, let's go through the steps to verify your solution.
First, you correctly expanded the numerator using the Maclaurin series for cosine:
1 - cos(x) = 1 - (1 - x^2/2! + x^4/4! - ...)
Next, you expanded the denominator using the Maclaurin series for exponential:
1 + x - e^x = 1 + x - (1 + x + x^2/2! + x^3/3! + x^4/4! + ...)
However, in your subsequent steps, it appears there may have been an error in the calculations. Let's correct it:
(1 - cos(x))/(1 + x - e^x) = (1 - (1 - x^2/2! + x^4/4! - ...))/(1 + x - (1 + x + x^2/2! + x^3/3! + x^4/4! + ...))
Simplifying the numerator, we have:
1 - (1 - x^2/2! + x^4/4! - ...) = x^2/2! - x^4/4! + ...
Now, let's simplify the denominator:
1 + x - (1 + x + x^2/2! + x^3/3! + x^4/4! + ...) = - (x^2/2 + x^3/3! + x^4/4! + ...)
Combining the numerator and denominator, we get:
(x^2/2! - x^4/4! + ...)/(-(x^2/2 + x^3/3! + x^4/4! + ...))
Now, we can cancel out the common terms of x^2/2!, -x^4/4!, and so on:
(x^2/2! - x^4/4! + ...)/(-(x^2/2 + x^3/3! + x^4/4! + ...)) = (x^2/2!)/(x^2/2) = 1
Therefore, the limit as x approaches 0 of (1 - cos(x))/(1 + x - e^x) is equal to 1, not -11/17 as you stated.
It's always a good idea to double-check your calculations and go through the steps again if you're unsure.