Finding limits at infinity in R - r

Question

I am doing an R code to evaluate limits.
I am not sure if my code even works I just run it and then it doesn't give anything and R stop debugging code / gets stuck, nothing works right after not even print statements. fn is supposed to be any function and tol is the error tolrence, I want to stop the program when the consective terms difference is less than 1e-6. I have to restart R and I always get the message "R session is currently busy" when I try to close R studio
lim<-function(funx,tol=1e-6){
n<-1
while(TRUE){
n<-n+1
term<-funx
next_term<-term+funx
if(abs(term-next_term)<tol){
break
}
}
return(term)
}
n<-1
fn<-(1/5)**n
lim(fn)

Answer 1

You made some mistakes in your program. For one, you always add the same number (funx) which will always be 0.20 and never smaller than the tolerance, so you get an endless loop.
If you want to call a function each time, you have to define this function and pass it to the lim() function. Otherwise, you just define fn as 0.20 and pass it as a double value to the function. It will never change.
If you want to find the limes of (1/5)^n, you can do it like that:
lim = function(f,x=1,tol=0.0001){
next.diff=tol
while(next.diff>=tol){
next.diff = abs(f(x)-f(x+1))
x = x + 1
}
return(list("Iterations"=x,"Limit"=f(x),"Next Value"=f(x+1)))
}
my.fun = function(x){(1/5)^x}
lim(my.fun,1,1e-6)
It wil lthen call the function for inceasing values of x and abort the loop as soon as the tolerance is reached. In this example:
> lim(my.fun,1,1e-6)
$Iterations
[1] 10
$Limit
[1] 1.024e-07
$`Next Value`
[1] 2.048e-08
So, at (1/5)^10 you already reach a value where the next iteration is closer than your tolerance. It's safe to say that it would converge to 0.
You can define any function of a value x and pass it to this lim function with a starting value for x and a tolerance level.
EDIT: For the limes of sqrt(x+1)-sqrt(x), you would just have to define a new function of x (or of n, if you wish) and pass it to lim():
> fun2 = function(x){sqrt(x+1)-sqrt(x)}
> lim(fun2,1,1e-6)
$Iterations
[1] 3969
$Limit
[1] 0.007936008
$`Next Value`
[1] 0.007935009

Answer 2

It's unclear as to what you really want to find out here, but as far as I understood, you want to see where the sequence (not a function) (and of the type a^n) converges. Well, if that is the case, then you need to change your code to something like this:
lim<-function(a,tol=1e-6)
{
n<-1
repeat
{
term<-a^n;next_term<-a^(n+1)
if(abs(term-next_term)<tol) break
n<-n+1
}
return(term)
}
Ok so here's what I did:
I assumed that the sequence you input is of the form a^n where a is a constant term, and n increase on the set of natural numbers
I defined the value of n initially inside the loop (why? cause I want to iterate over all the possible values of n, one-by-one)
Then I defined the first term of the sequence (named as term). As assumed, it's a^n initially. So the next term (a.k.a. next_term in my code) should be a^(n+1).
Now take their absolute difference. If it satisfies the condition, break out from the loop. Else, increase the value of n by 1 and let the loop run once again.
Then finally, return the value of the term. That's all...
I hope you will now be able to understand where you went wrong. Your approach was similar, but the code was of something else.
Remember, in this code, you don't need to enter the value of n separately while calling the function.
Here's what it returned:
> lim(1/5)
[1] 5.12e-07
> fn<-1/12
> lim(fn)
[1] 3.34898e-07