ENGINEERING MATHEMATICS I (BWM 10103) Seksyen 2

PERSONAL DETAILS Name: Dr. Fariza Mohamad Room: Block C15, Room 17 Email: farizamd@uthm.edu.my

LIMITS AND CONTINUITY

CONTENTS Definition of limits Evaluation of limits Right sided limit Left sided limit Two-sided limit 3. Infinite limit 4. Technique of evaluating limits a) Limit at a point b) Limit at infinity .

CONCEPT OF LIMITS The limit of a function refers to behavior of a function as the independent variable approaches a given value. If the limit of function f (x) as x approaches the point a is the value L, then it is denoted as In simpler terms, the definition says that as x get closer and closer to a then f (x) must getting closer and closer to L. * Limit can be establish from tables and graphs. .

CONCEPT OF LIMITS Using tables Estimate the value of the following limit. 1. Choose values of x that get closer to closer to 2 2. substitute the values into . Doing this, gives the following table: x 1.5 1.9 1.999 1.9999 1.99999 2 2.00001 2.0001 2.001 2.01 2.5 f (x) 5.0 4.158 4.002 4.0002 4.00002 3.99999 3.9999 3.999 3.985 3.4 Approach to 2 from the left side Approach to 2 from the right side

CONCEPT OF LIMITS The function is going to 4 as x approaches 2, then Important !! Using tables of values to guess the value of limits is simply not a good way to get the value of a limit but they can help us get a better understanding of what limits are.

CONCEPT OF LIMITS One-sided limits: Right side Only look for limit at right side of the point. Right-handed x approaches a from right hand side. We write as .

CONCEPT OF LIMITS One-sided limits : Left side Only look for limit at left side of the point. Left-handed x approaches a from left hand side. We write as .

CONCEPT OF LIMITS Two-sided limits The limit is called two-sided limit. >> Requires value of limits from right and left sides. Two-sided limit of f (x) only exist if Then, Example: One and Two-sided limits

CONCEPT OF LIMITS Example: One and Two-sided limits

INFINITE LIMITS Infinite limits Infinite limits occur when and . All limits that have infinite limit = limit does not exist. Example: Infinite limits

TECHNIQUES OF COMPUTING LIMITS limits at a point Consider first limits at a point which is “limits of f (x) when x approaches any point a”. Some basic properties of limits Let a and k is any real number

TECHNIQUES OF COMPUTING LIMITS limits at a point

TECHNIQUES OF COMPUTING LIMITS limits at a point We will discuss algebraic techniques for computing limits of:- Polynomial functions Rational functions Radical functions, (function involve ) Piecewise functions

TECHNIQUES OF COMPUTING LIMITS limits at a point Polynomial functions Let f (x) is polynomial function Then, Example: Polynomial function

TECHNIQUES OF COMPUTING LIMITS limits at a point Polynomial example Evaluate the following: a) b)

TECHNIQUES OF COMPUTING LIMITS limits at a point Rational functions Let f (x) is rational function , with d(x) and n(x) polynomial function Then, ,provided that . But how if ??? does not exist. 1 If 2 If , we have to factor d(x) and n(x) (by common factor (x – a) ) then cancel out (SIMPLIFIED THE FUNCTION).

TECHNIQUES OF COMPUTING LIMITS limits at a point Rational functions Evaluate the following:

TECHNIQUES OF COMPUTING LIMITS limits at a point Radical functions, (function involve ) Let f (x) is radical function , with e(x) and m(x) radical function Then, For Rationalize is:- Rationalize??? ,provided that . But how if ??? does not exist. 1 If 2 If , rationalize e (x) or m (x)

TECHNIQUES OF COMPUTING LIMITS limits at a point Radical functions, (function involve ) Example: Radical function

TECHNIQUES OF COMPUTING LIMITS limits at a point Piecewise functions Piecewise function is defined by The limits of piecewise function is obtained by finding the one-sided limits first.

TECHNIQUES OF COMPUTING LIMITS limits at a point Piecewise functions Example: Piecewise function

4. Technique of evaluating limits a) Limit at a point TECHNIQUES OF COMPUTING LIMITS limits at a point Summary: Definition of limits Evaluation of limits Right sided limit Left sided limit Double sided limit 3. Infinite limit 4. Technique of evaluating limits a) Limit at a point

Lecture 05.03.2013

TECHNIQUES OF COMPUTING LIMITS limits at infinity Limits of f (x) when x increase without bound ( ) or x decrease without bound ( ). Some basic properties of limits Let k is any real number

TECHNIQUES OF COMPUTING LIMITS limits at infinity

TECHNIQUES OF COMPUTING LIMITS limits at infinity We will discuss algebraic techniques for computing limits of:- Polynomial functions Rational functions Radical functions, (function involve )

TECHNIQUES OF COMPUTING LIMITS limits at infinity Polynomial functions Let f (x) is polynomial function Then, depends on the highest degree term and consider

TECHNIQUES OF COMPUTING LIMITS limits at infinity Polynomial functions Example: Polynomial function

TECHNIQUES OF COMPUTING LIMITS limits at infinity Rational functions Let f (x) is rational function Then, Step 1 Divide each term in n(x) and d(x) with the highest power of x. Step 2 Substitute to find the answer

TECHNIQUES OF COMPUTING LIMITS limits at infinity Rational functions Example: Rational function

TECHNIQUES OF COMPUTING LIMITS limits at infinity Radical functions, (function involve ) Let f (x) is radical function Try substitute the value of into the function. If can’t find the answer, consider these cases: Case 1 Case 2 f (x) = m (x) only Rationalize m (x) Step 1 Divide m(x) and e(x) with the highest power of x Step 2 Substitute to find the answer Special case? Use concept

TECHNIQUES OF COMPUTING LIMITS limits at infinity Radical functions, (function involve ) Example: Radical function (d)

CONTINUITY We perceive the path of moving object as an unbroken curve without gaps, breaks or holes. In this section, we translate the unbroken curve into a precise mathematical formulation called continuity. A function f (x) is said to be continuous at x = a if Provided that EXIST and DEFINED.