Suppose you have a rational function that its denominator is (X-c) and numerator can be any function. Suppose you need to find the limit of this rational function given as x goes to value c. Can you always say that for this problem the limit does not exist?

No, we cannot always say that the limit of a rational function does not exist when the denominator is of the form (x-c), where c is a constant value.

In fact, if the numerator is also defined at x=c, the limit of the rational function as x approaches c can exist and be evaluated. To find the limit, we can use the concept of continuity.

To evaluate the limit of a rational function, we substitute the value of the variable (in this case, x=c) directly into the function. However, if this substitution leads to an undefined expression (such as division by zero), then the limit does not exist.

So, if the function is defined at x=c (i.e., the numerator is also defined at x=c) and if substituting x=c does not result in an undefined expression, then the limit of the rational function as x approaches c exists and can be found.

If the function is not defined at x=c, or if substituting x=c leads to an undefined expression, then the limit does not exist. This could occur when the numerator is also zero at x=c, resulting in the infamous "0/0" indeterminate form.

Therefore, the existence of a limit for a rational function with a denominator of the form (x-c) depends on the specific behavior of the numerator as x approaches c.