Calculus 1 NotesSection 1.2Page 1 of 5
Section 1.2: The Concept of Limit
Big Idea:A limit is the number to which a sequence of ever-better approximations converge.
For example, in section 1.1 you used secant lines to approximate the tangent line at a given point on the graph of a function in order to calculate the slope of the tangent line.
One hard part about limits is finding a function that accurately describes the ever-better approximations. In this section, however, the functions will be given to us. Another hard part about limits is rigorously proving what the limit actually is; we’ll deal with that in section 1.6.
Big Skill: You should be able to use graphical and numerical evidence to conjecture the limit of a given function at a given point.
Recall that yesterday, we learned an approximation technique to find the slope of the tangent line at a point on the curve. The approximation was to pick points near the given point, and calculate the slope of the secant line. As the chosen points got closer and closer to the given point, the calculated slopes seemed to converge to a limit.
Tangent line to the parabola at the point (2, 5) and an approximating secant line through (1, 2).
How can we find the limit of the secant line slopes exactly? We start by writing a function that calculates the secant line slope for any point on the curve that we pick. Any point on the curve (x, y) really has coordinates (x, x2 + 1) since . So, the slope of the secant line is:
Now we are going to look at the value of msec(x) as x approaches 2. At first glance, that appears impossible to do, since we can’t plug x = 2 into the function, because that would result in division by 0. So, let’s examine numerical and graphical evidence to find the limit of this function as x approaches 2.
- Numerical approximations for the function msec(x) as x approaches 2:
- Graph of the function msec(x) as x approaches 2:
- What does the limit of the function msec(x) seem to be as x approaches 2?
- Why does this function have a well-defined limit as x approaches 2?
- Notice what we have done here: we have created a new function, msec(x), that tells us something about the function f(x) = x2 + 1.
Practice:
Write a function that computes the slope of the secant lines for the graph of y = x3 near the point (1, 1). Use numerical and graphical evidence to conjecture the limit of that secant slope function. This conjectured limit is the slope of the tangent line to y = x3 at the point (1, 1).
To be honest, we have exceeded the scope of this section already. In this section, we will not have to create our own secant line slope functions; the book simply will give us a function, and ask us to conjecture the limit of that function at a given x value using graphical and numerical evidence. For example:
Practice:
Use numerical and graphical evidence to conjecture the limit of the function as x approaches 2.
Some formalized language of limits:
One-sided limit: the number to which a sequence of values of a function evaluated at x values on one side or the other of an x value of interest converge.
Examples of writing one-sided limits:
- For the function , “the limit of f(x) as x approaches 2 from the left is equal to 4” is written as:
- For the function , “the limit of f(x) as x approaches 2 from the right is equal to 4” is written as:
- For the function , “the limit of g(x) as x approaches 2 from the left is undefined” is written as:
- For the function , “the limit of g(x) as x approaches 2 from the right is undefined” is written as:
A limit exists if and only if (iff) both corresponding one-sided limits exist and are equal.
, for some number L, if and only if
In this section, you will be expected to conjecture the limit of a given function at a given x value by:
- examining the values of the function forx values on either side of the given x value (i.e., looking at one-sided limits).
- examining the graph of the function near the given x value.
Practice: