Limits - Muneera Albahar

My name is Professor Muneera Albahar and today I am going to teach you the concept of limits.

Learning Objectives:

1.     Understand the meaning of limits and why we study it.

2.     Learn how to calculate the limits for a specified function. (Students should be able to explain how they to come to a particular solution.)

3.     Real World Application

Learning Materials:

Youtube Video Link: http://www.youtube.com/watch?v=skM37PcZmWE

Reference Book: “Fundamentals of University Mathematics” by C M McGregor, John Nimmo, Wilson Stothers.

Website: http://www.sparknotes.com/math/calcab/functionslimitscontinuity/section2.rhtml

       http://www.mathsisfun.com/calculus/limits.html

Lesson Outline

Ø  Why we need to learn limits?

Limits are the most fundamental concept in Higher Mathematics, especially Mathematical Analysis. Most concepts, such as continuity, differential, integration, are defined on the basis of limits. For example, the definition of function continuity at a certain point is always written as the following:

“In the function f, at the certain point x₀ of its domain, if the limits of f(x) from left hand side and right hand side as x₀ approaches f(x₀) through the domain of f exist and both are equal to f(c)”.

To simplify, a limit is our best prediction of a point that we are unable to observe, which means that we can learn a lot about the function using limits. For example, we can check what happens to the function at a discontinuity by taking a limit at this point. Graphically, it also tells us that the function approaches either positive or negative infinity.  

Ø  Definition

In the 19^th century, Augustin-Louis Cauchy, followed by Karl Weierstrass, formalized the definition of the limit, which became widely known as the (ε, δ)-definition of limit.

Suppose f is a real-valued function, and c is a certain number of its domain. The expression  means when the values of x whose distance from c is less than some positive number δ, that is, values of x within either (c − δ, c) or (c, c + δ), there is a certain positive number ε which represent any small positive number, that f(x) certainly lies in the interval (L-ε, L+ε).

Ø  Graph Explanation

Whenever a point x is within in δ units of c, f(x) is within ε units of L.

Ø  Rules For Computing Limits

Three fundamental Rules:

1.     The limit of a constant will simply be the constant

2.     If two sequences have limits that exist, then the limit of the product is the product of the limits.

3.     If two sequences have limits that exist, then the limit of the sum of sequences is the sum of the limits of the sequences.

If our sequence is described by a fraction with powers of n on the top and the bottom, we may follow the following rules:

1.     If the degree of n is higher on top, then the limit is infinity.

2.     If the degree of n is higher on the bottom, then the limit is zero.

3.     If the degree of n is the same in the numerator and denominator, then the limit is the ratio of the leading coeffecients.

Ø  Example:

Example1.

The definition of the derivative of a function at a certain point uses the concept of limits.

Suppose f is a real-valued function, and x₀ is a certain point of its domain in the f. The derivative f’(x₀) f’(x₀) = lim x

Example2.

Find f(x)=

Note: the point x=3 for the function  is meaningless since the denominator cannot be zero, so we should use the limit to predict the value at x=3.

Solution: f(x)==

Ø  Real World Application:

Suppose if I keep tossing a coin as long as it takes, how likely am I to never toss a head?

The possibility of never tossing a head is decreasing. Within the frequency increasing, the limit of the possibility is 0.

The mathematical answer to this is:

P(N)=( )^N

lim_N→∞p(N)=0




Class dismissed!