Properties.Methods and Procedures to calculate limits and continuity

Sometimes we seem lost through the details when studying a subject. However if we get the big picture it becomes easy to continue studying.  A math topic is structured in concepts, rules or properties and theorems. This is the theoretical part. Then come the applications. The theories are applied in the applications but the procedures and methods are mastered through practice. Knowing some key theories and procedures can help tremendously in the solutions of problems. In this post I will highlight the properties of limits and continuity, the methods and procedures to solve problems.

Properties of limits

The properties of limit show how to calculate the limits of a combination of functions like the sum, the difference, the multiplication and division of functions. It shows also how to calculate the limit of the square root of a function.

Properties of continuous functions

Methods for determining limits

There are three methods that can be used to determine a limit. These methods are: graph, table and algebra. The graph method consists in determining a limit from the graph. The table method consists in calculating the limit to the left and to the right by drawing a table for each one-sided limit. The table allows to see the behavior of the values of f(x) as x gets closer and closer to a fixed value. From there we can conclude if the limit to the right or to the left exists. If the limits from both sides exist and are equal then the limit of the function exists at the given value. The algebra method consists by substituting the value of the independent variable in the function.

Method for determining if a function is continuous

To determine if a function is continuous, we find out if it satisfies the three following conditions:
1) It is defined at a specified point "a"
2) The limit at the point "a" exists
3) The limit of the function at the point "a" is equal to f(a).

If you are interested in learning more about these concepts you can subscribe to this free Introductory Calculus course or this complete course Calculus AB

Limits and Continuity vocabulary

These definitions can be best learned by watching some videos and observing the graphs of the functions. If you have learned the previous lessons there shouldn't be any problems mastering them

Limit

If the values of a function f approach a number L as the variable gets closer and closer to a number "a", then L is said to be the limit of the function f at the poin "a".

Two-sided limit

A two-sided limit is a limit where both the limit to the left and the limit to the right are equal

One-sided limit

A one-sided limit is a limit taken as independent variable approaches a specific value from one side (from the left or from the right).

Limit to the left

If the values of a function approach a number L as the independent variable gets closer and closer to a number "a"in the left direction. then the number L is said to be the limit of the function f to the left at the point "a"

Limit to the right

If the values of a function approach a number L as the independent variable gets closer and closer to a number in the right direction, then the number L is said to be the limit of the function f to the right at the point "a".  

Asymptote

An asymptote is a straight line to a curve such that as a point moves along an infinite branch of a curve the distance from the point to the line approaches zero as and the slope of the curve at the point approaches the slope of the line 

Vertical asymptote

A vertical asymptote is a vertical line to a curve such that as a point moves along an infinite branch of the curve the distance from the point to the line approaches zero

Horizontal asymptote

A horizontal asymptote is a horizontal line to a curve such that as a point moves along an infinite branch of the curve the distance from the point to the line approaches zero.  

End behavior

This is the behavior of the arm branches or infinite arm branches of a curve. In the case of a curve with a vertical asymptote the arm branch approaches the asymptote more and more.

Continuity of a function at a point

A function f is continuous at a point "a" if the limit of the function when x approaches "a" is equal to the value of the function at this point 

Continuity of a function on an interval

A function f is continuous on an interval if it is continuous at every point of the interval

Continuity of a function to the left at a point

A function f is continuous to the left at a point "a" if its limit to the left is equal to the value of the function at this point

Continuity to the right

A function f is continuous to the right at a point "a" if its limit to the right is equal to the value of the function at this function.

Continuous function

A continuous function is a function of which the graph can be drawn without lifting the pencil. Its graph has no hole, jump or asymptote. Algebraically a function f is continuous if for every value of its domain the limit exists.

Discontinuous function

A discontinuous function is a function of which the graph has hole, jump or asymptote. Algebraically a discontinuous function is either not defined at a point of its domain, doesn't have a limit at this point or the limit at this point is not equal to the value of the function at this point.

Removable discontinuity

Graphically a removable discontinuity is a hole in a graph or a point at which the graph is not connected there. The graph can be connected by filling in the single point.
Algebraically a removable discontinuity is one in which the limit of the function does not equal to the value of the function. This may be because the function does not exist at that point.

Non-removable discontinuity

A non-removable discontinuity is a point at which a function is not continuous or is undefined. and cannot be made continuous by giving a new value at the point. A vertical asymptote and a jump are examples of non-removable discontinuity.

Intermediate value theorem

If a function f is continuous over an interval [a b] and V any number between f(a) and f(b), then there is a number c between a and b such as f(c) = V (that is f is taking any number between f(a) and f(b)). We can deduce from this theorem that if f(a) and f(b) have opposite signs, there is a number c such as f(c) - 0. This can be used to find the roots of a function,

If you are interested in learning more about these concepts you can subscribe to this free Introductory Calculus course or this complete course Calculus AB