Convergence of a function refers to the behavior of the values of the function as the input approaches a specific value or goes towards infinity. It describes whether the function approaches a certain value or diverges as the input gets closer to a particular value.
To determine the values at which a function converges, we need to analyze the behavior of the function as the input approaches a specific value or goes towards infinity. This can be done by evaluating the limit of the function as the input approaches the given value or infinity.
Pointwise convergence of a function means that for each input, the values of the function approach a specific value. On the other hand, uniform convergence means that the values of the function approach a specific value at the same rate for all inputs.
To prove that a function converges at a certain value, we need to show that the limit of the function as the input approaches the given value exists and is equal to the value itself. This can be done by using the definition of limits and evaluating the limit of the function.
Yes, there are certain cases where a function may not converge at a certain value. This can happen when the limit of the function does not exist or when the function oscillates between different values as the input approaches the given value. In these cases, the function is said to diverge.