The notion of limit a function at one point
a function y = f (x) may not be defined for a certain point, say x = XO, as it happens with y = log x at point x = 0, or as it happens with y = TG x at point x = P/2. In fact, a function y = f(x) can show a strange behavior at a certain point x = xo. To better understand these possible anomalies of some functions the notion of limit of a function is introduced at one point.
There are functions that at a certain point x = XO have a discontinuity, suffering their graph of a “jump” as shown in the figure below.
Sometimes when calculating limits, we find certain expressions whose values we do not know a priori. These are the indetermination. For some of them, there are rules that allow us to calculate their value (as in the case of 1 ∞). But most of the indetermination is not resolved in such a direct way, but we must perform a series of operations or calculations to determine their values.
We must say that in fact, the differential calculus provides us with an advantageous and straightforward method under certain conditions: the rule of Hôpital. But we will not use this rule as we have a section specially dedicated to it: limits by Hôpital.
Let’s look at the indetermination mentioned above and some of the procedures that can be used to resolve them.
Left and right limits:
Just to remember: the criteria for the existence of a boundary are.
- That there is the limit of f(x) when x tends to “C” on the Left
- That there is the limit of f(x) when x tends to “C” on the right
- That the boundary on the left is equal to the limit to the right
The value x = c is any actual number. Remember also that a limit is always a value of “Y”, not “X”.
In addition, there are 3 continuity criteria for a function in x = “C”. An f (x) function is continuous in the value x = c when it complies:
- The function is defined for f (c), so F (c) exists.
- There is the limit of f(x) when x tends to C (that is to meet the 3 criteria of the existence of a limit mentioned before).
- The value of f (c) is equal to the value of the límtie of f(x) when x tends to “C”.