A number of examples of how to find limits.
If a function is know to be continuous at , then by the definition of continuity we have, .
We will take is as known that the elementary functions – polynomial, rational (polynomial/polynomial), trigonometrical, exponential, logarithmic) – are continuous everywhere.
Therefore, although not very exciting, we are able to say such things as:
A very useful rule (it’s a theorem really) for finding limits is L’Hôpitals rule (LH), which is really a collection of rules. The most comprehensive form of the rule is as follows:
Suppose you wish to find something like where [ ] may be .
Suppose further that where either
– ( type), or
– ( type).
Then the LH rule says look at and if you can do it, say find it to be ( ) then the original limit is the same.
Danger: Do not use LH when it is not applicable.
For example is not the same as .
The LH rule gives us with no effort some useful results. For example:
Let a be a point in the domain of a function. The function is said to be continuous at if .
Otherwise f is said to be discontinuous at . The function may be discontinuous at a for various reasons:
i) may not exist, may not exist, or both may not exist.
ii) both may exist, but may not be equal.
iii) both may exist, and be equal, but not be equal to f(a).
If f is continuous at all points of its domain it is simply said to be continuous. (Sometimes for emphasis mathematicians say f is continuous everywhere).
Note that it is a silly question to ask whether or not a function is continuous at a point which is not in the domain. For example is continuous at x=1?
Consider the function
As x gets closer to 2 from above we notice that f(x) gets as close as we like to 1. We say that f has limit 1 as it tends to 2 from above (some people say from the right).
We write or, as .
Note that the value of f at x=2 is not relevant, for the concept of limits we do not even need x=2 to be in the domain for f.
In the same spirit,
Consider the function
As we get closer to 0 from above f(x) gets as large as you like, we say f(x) tends towards infinity.
We write, , or
It does not mean that f(x) gets close to infinity. The value of f at x=0 is again irrelevant.
In the same spirit we have, .
If for a function f(x) and a point x=a we have
then we say that f has a limit h as x tends to a, and we write or as .
Likewise if we write simply .
For example, .
Consider the function .
As x gets large and positive we see that f(x) gets as close to 2 as we like.
We write or as .
Likewise or as .
A function is said to be an even function if,
and to be an odd function if,
Note that these definitions require D(f) to lie symmetrically about x=0 (ie ). Clearly even functions have graphs that are symmetrical about the y-axis, whilst for odd functions the origin is a centre of rotational symmetry.
Examples are for even, and for odd. Most functions are neither odd nor even. However every function (subject to having a symmetrical domain) may be expressed as the sum of an odd and even function since , which is even + odd.
ie. even + odd.
Let f be a function and let . We often call x the independent variable and y the dependent variable. Now consider specifying a number and working out what x’s are in the domain of the f, for which . There may of course be many such x’s corresponding to a particular y, but functions for which there is only ever one such x have a special name, they are called injections (their old name was one-to-one).
A function is an injection if for all x,y
Or if you prefer
( – for all and – implies.)
Injections are easily recognised from their graphs in that any line parallel to the x axis will meet the graph in at most one point.
Let f be an injection, then we use to denote that single element which is such that .
The function is called the inverse function of the f.
By the definition of we have, .
It is also clear that .
Note that functions that are not injections do not have inverse functions.
Find the inverse function of the function f given by:
Solution: To have a sketch is a great help
Since the rule for f is given in two pieces we find the rule for in two pieces.
Whence is given by
Note that f is an injection and so exists then the graphs f and will be reflections of one another in the line y = x.