DifferentialCalculusNotes

Informal definition of limit:

Suppose ƒ(x) is a real-valued function and c is a real number. The expression:

$\lim_{x \to c}f(x) = L$

means that ƒ(x) can be made to be as close to L as desired by making x sufficiently close to c. In that case, we say that "the limit of ƒ of x, as x approaches c, is L".

Precise definition:
DEFINITION: The statement $tex2html_wrap_inline83$ has the following precise definition. Given any real number $tex2html_wrap_inline85$ , there exists another real number $tex2html_wrap_inline87$ so that

if $tex2html_wrap_inline89$ , then $tex2html_wrap_inline91$ .

Basic Limits

If c is a constant, then $the limit as x goes to a of c is c$ .

$the limit as x goes to a of x is a$ .

Limit Laws

Addition Law

If the limits $the limit as x goes to a of f(x)$ and $the limit as x goes to a of g(x)$ both exist, then

$the limit as x goes to a of f(x)+g(x) equals the limit as x goes to a of f(x) + the limit as x goes to a of g(x)$ .

Subtraction Law

If the limits $the limit as x goes to a of f(x)$ and $the limit as x goes to a of g(x)$ both exist, then

$the limit as x goes to a of f(x)-g(x) equals the limit as x goes to a of f(x) - the limit as x goes to a of g(x)$ .

Constant Law

If c is a constant, and the limit $the limit as x goes to a of f(x)$ exists, then

$the limit as x goes to a of c*f(x) = c* the limit as x goes to a of f(x)$ .

Multiplication Law

If the limits

$the limit as x goes to a of f(x)$

and

$the limit as x goes to a of g(x)$

both exist, then

$the limit as x goes to a of f(x)*g(x) = the limit as x goes to a of f(x) * the limit as x goes to a of g(x)$

Division Law

If the limits $the limit as x goes to a of f(x)$ and $the limit as x goes to a of g(x)$ both exist, and $the limit as x goes to a of g(x) is not zero$ , then

$the limit as x goes to a of (f(x)/g(x)) = (the limit as x goes to a of f(x))/(the limit as x goes to a of g(x))$ .

Power Law

If n is an integer, and the limit $the limit as x goes to a of f(x)$ exists, then

$the limit as x goes to a of (f(x))^n = (the limit as x goes to a of f(x))^n$ .

Root Law

If n is an integer, the limit $the limit as x goes to a of f(x)$ exists, and that limit is positive if n is even, then

$the limit as x goes to a of the nth root of f(x) = the nth root of the limit as x goes to a of f(x)$ .

Squeeze Law

If $f(x) <= g(x) <= h(x)$ for all x in an open interval that contains a, except possibly at a itself, and $the limit as x goes to a of f(x) = the limit as x goes to a of h(x) = L$ , then

$the limit as x goes to a of g(x) = L$ .

Composition Law

If f is continuous at b and $the limit as x goes to a of g(x) = b$ , then

$the limit as x goes to a of f(g(x)) = f(b) = f(the limit as x goes to a of g(x))$ .

You can find an Excel file with examples of calculating limits here

One Sided Limits

Right-handed limit

We say

provided we can make f(x) as close to L as we want for all x sufficiently close to a and x>a without actually letting x be a.

Left-handed limit

We say

provided we can make f(x) as close to L as we want for all x sufficiently close to a and x<a without actually letting x be a.

Example Find the value of the following limits.

Solution

here’s the graph.

$TheLimit_G4$

So, we can see that if we stay to the right of (i.e. ) then the function is moving in towards a value of 1 as we get closer and closer to , but staying to the right. We can therefore say that the right-handed limit is,

Likewise, if we stay to the left of (i.e ) the function is moving in towards a value of 0 as we get closer and closer to , but staying to the left. Therefore the left-handed limit is,

In this example we do get one-sided limits even though the normal limit itself doesn’t exist.

Example Find the value of the following limits.

Solution

The graph of this function looks like:

$TheLimit_G2$

In this case regardless of which side of we are on the function is always approaching a value of 4 and so we get,

Given a function f(x) if,

then the normal limit will exist and

Likewise, if

then,

Limit of a function at infinity

A related concept to limits as x approaches some finite number is the limit as x approaches positive or negative infinity. This does not literally mean that the difference between x and infinity becomes small, since infinity is not a real number; rather, it means that x either grows without bound positively (positive infinity) or grows without bound negatively (negative infinity).

For example, consider $f(x) = {2x-1 \over x}$

f(100) = 1.9900
f(1000) = 1.9990
f(10000) = 1.9999

As x becomes extremely large, the value of f(x) approaches 2, and the value of f(x) can be made as close to 2 as one could wish just by picking x sufficiently large. In this case, we say that the limit of f(x) as x approaches infinity is 2. In mathematical notation,

$\lim_{x \to \infty} f(x) = 2.$

For example, consider $f(x) = {2x-1 \over x}$

f(100) = 1.9900
f(1000) = 1.9990
f(10000) = 1.9999

$\lim_{x \to \infty} f(x) = 2.$

Infinite Limits

We say

if we can make f(x) arbitrarily large for all x sufficiently close to x=a, from both sides, without actually letting

We say

if we can make f(x) arbitrarily large and negative for all x sufficiently close to x=a, from both sides, without actually letting ,

Examples Ex

Here is a sketch of the graph.

$InfiniteLimits_G1$

Continuity

Suppose we have a function that maps real numbers to real numbers and whose domain is some interval, like the functions h and M above. Such a function can be represented by a graph in the Cartesian plane; the function is continuous if, roughly speaking, the graph is a single unbroken curve with no "holes" or "jumps".

In general, we say that the function f is continuous at some point c of its domain if, and only if, the following holds:

The limit of f(x) as x approaches c through domain of f does exist and is equal to f(c); in mathematical notation, $\lim_{x \to c}{f(x)} = f(c)$ . If the point c in the domain of f is not a limit point of the domain, then this condition is vacuously true, since x cannot approach c through values not equal c. Thus, for example, every function whose domain is the set of all integers is continuous.

We call a function continuous, if, and only if, it is continuous at every point of its domain. More generally, we say that a function is continuous on some subset of its domain if it is continuous at every point of that subset.

Example Given the graph of f(x), shown below, determine if f(x) is continuous at , and .

$Continuity_G1$

Solution

To answer the question for each point we’ll need to get both the limit at that point and the function value at that point. If they are equal the function is continuous at that point and if they aren’t equal the function isn’t continuous at that point.

First

The function value and the limit aren’t the same and so the function is not continuous at this point. This kind of discontinuity in a graph is called a jump discontinuity. Jump discontinuities occur where the graph has a break in it is as this graph does.

Now .

The function is continuous at this point since the function and limit have the same value.

Finally

The function is not continuous at this point. This kind of discontinuity is called a removable discontinuity. Removable discontinuities are those where there is a hole in the graph as there is in this case.

For a proof that Lim_θ−>0 (sin θ)/θ = 1, see here:PDF

An excel file with computations investigating derivatives of exponential functions can be found here

Basic Limits

Limit Laws

Limit of a function at infinity

Infinite Limits

For a proof that Limθ−>0 (sin θ)/θ = 1, see here:PDF

For a proof that Lim_θ−>0 (sin θ)/θ = 1, see here:PDF