C++ Endless loop with function sin x problem

I am practicing for my final exam and professor sent us a last year's copy and in it, there is this assignment:

I made a code for it, but it seems it doesn't work properly, it ends up in an endless loop most of the times. This is my code:

int main(){
    double x;
    double sinx;
    int n=1;
    int sign=-1;
    cin>>x;
    sinx=0;
    for(;;) {
        int fact=1;
        for(int i=1;i<=2*n+1;i++){
            fact*=i;
        }
        double term=pow(x,2*n+1)/fact;
        if(term==0){
            break;
        }
        sinx += term * sign;
        sign *= -1;
        n++;

    }
    cout<<sinx;
}

I really tried everything.

EDIT:

The bug is in integer overflow of fact. There should be a check that integer overflow hasn't happened.

#include<cassert>
#include<iostream>
#include<cmath>

int main(){
    double x;
    double sinx;
    int n=1;
    int sign=-1;
    std::cin>>x;
    sinx=0;
    bool overflow = false;

    for(;;) {
        int64_t fact=1;
        for(int i=1;i<=2*n+1;i++){
            old_fact = fact;
            fact = fact * i;
            // check for overflow
            if (fact / i != old_fact) {
               overflow = true;
               break;
            }
        }
        if (overflow)
           break;


        double term=std::pow(x,2*n+1)/fact;
        if(term==0){
            break;
        }
        std::cout<<"n=" << n << ", fact = " << fact << ", term: " << term <<"\n";
        sinx += term * sign;
        sign *= -1;
        n++;
        std::cout<<"sinx=" << sinx<<"\n\n";
    }
    std::cout<<sinx;
}

godblot link: https://godbolt.org/z/zP8sT6ovq

Old answer:

The problem is with checking if a floating point number is exactly equal to zero.

        if(term==0){

This may not ever evaluate to true. It may keep oscillating around a value close to zero. What you could do is to compare if term is arbitrarily close to zero. You can hard code a value like 0.000001 (which I recommend for a class project) or you can use epsilon if you want to get more technical.

So the code will be like:

if (std::abs(term) < 0.000001) {
  break;