1.6b Limits at Infinity
In this lesson, we will explore limits at infinity. These describe the end behavior of the function, or in
other words, how the function behaves as x gets very large in magnitude. Lets begin with an example.
Suppose you put a warm
2.2a The Derivative as a Function and
Graphing the Derivative
In this module, we will examine the derivative as a function in its own right. We will also calculate more
derivatives, and see how to use the graph of a function to sketch the graph of its der
2.2b Properties of the Derivative
In this lesson, we will discuss differentiable functions. A function is differentiable at a point if it has a
derivative there. An obvious question to ask is the following: when does a function fail to be
differentiable?
2.1a Definition of the Derivative
In this lesson, we introduce a core concept of calculus, the derivative. The derivative measures the
instantaneous rate of change of a function at a particular input. Lets start with a thought experiment.
Ill bet you $2 I
1.5b The Intermediate Value Theorem
In this lesson we introduce an important theorem in calculus, the Intermediate Value Theorem. This is
important in itself, but also because it is an example of what we call an existence theorem in
mathematics. These tel
1.5a Continuity
In this lesson, we introduce the concept of continuity. This concept puts a mathematical framework on
the idea of a functions values being connected to one another. This is another of those mathematical
concepts which is best approached gr
1.6a Infinite Limits
In this lesson, we will concentrate on those types of limits whose values are infinite. We will also discuss
a nice geometric feature of the graph of a function, called a vertical asymptote, which arises when a
function has an infinit
1.3a Definition of the Limit
In this lesson, we are going to introduce a core mathematical concept which, when added to the
machinery of algebra, really makes calculus the powerful set of tools that it is. This idea is that of the
limit of a function. Let
1.4b Trigonometric Limits and the Squeeze Theorem
Previously, we have seen how to evaluate limits graphically, numerically, and through the use of limit
laws. In this lesson, we will extend our knowledge of limits into the realm of trigonometric functions
1.4a Calculating Limits
In this lesson we will talk about limit laws. A limit computation can be just a plain application of the limit
laws but sometimes it might require a bit more work since along the way of computing a limit we might
feel like we are a
1.3b What Can Go Wrong with Limits
In this lesson we will explore different situations to demonstrate things that can go wrong when
computing the limits. You should be familiar with the fact that for the limit to exist the left and right
limits should exi
Calculus I (Math-UA-121.036)
Fall 2016-HW 3
Section 1.6
1. Evaluate the following limits.
(a) lim
x
(b) lim
x
cos x
x 2x
1
(d) lim x sin
x
x
4x 3
25x2 + 4x
4x4 + 9x 2x2
(c) lim
2. Sketch the graph of an example of a function f that satisfies all of the
Calculus I (Math-UA-121.036)
Fall 2016
Section 1.1
1. Find the domain and range of the following functions:
(a) f (x) = 4 x4
(b) f (x) = ln(x2 1) + 5
(c) f (x) =
x2 1
x+1
2. Determine whether the given function f is even, odd, or neither. Provide a comple