tiny-tim said:Hi Jamin2112!
If the limit exists, and is x, then you can substitute x for xn and xn+1 in the same equation.
A limit exists when the values of a function approach a certain value as the input approaches a specific value, without ever reaching that specific value.
To prove that a limit exists, you must show that the values of the function get closer and closer to a single value as the input gets closer to a specific value. This can be done through various methods such as using the formal definition of a limit, graphing the function, or using algebraic techniques.
The formal definition of a limit states that the limit of a function at a specific point is the value that the function approaches as the input approaches that specific point. This can be written as: lim x->a f(x) = L, where L is the limit and a is the point the input is approaching.
Yes, a limit can exist even if the function is not defined at that point. This is because the limit is based on the behavior of the function as the input approaches a specific value, not the actual value at that point.
Some common techniques for finding limits include using algebraic manipulation, factoring, and using limit laws (such as the sum, difference, and product laws). Additionally, graphing the function or using the formal definition of a limit can also be helpful in finding limits.