Findinglimitslefttoright

//**FINDING LIMITS FROM LEFT TO RIGHT**// **Continuous Functions** The function f(x) is ** continuous at x = a **if The function f is said to be ** continuous on its domain ** if it is continuous at each point in its domain. If f is not continuous at a particular a, we say that f is ** discontinuous **at a or that f has a ** discontinuity ** at a. 6x - 2 || The functions whose graphs are shown below are said to be continuous since these graphs have no "breaks", "gaps" or "holes".
 * x[[image:http://people.hofstra.edu/stefan_waner/realworld/lim.gif]]a || f(x) exists, and equals f(a). ||
 * 3x2 - x +1, ||
 * (x2 - 1)1/2
 * e(x2 - 1)1/2/x ||
 * (log3(4x2 - ex))2/3 ||

[|Continuous function video] Given a function f(x) and objective a, the left-hand limit of f(x) as x approaches a has the value of L if the values of f(x) get closer and closer to L as for values of x are to the left of a but increasingly near to a.

The notation used is lim f(x) (left hand limit) and lim f(x) (right-hand limit)x--->a- x->a+

The Difference Between Left and Right handed Limits

This function has a negative sign after the limit 1 of x. This means the limit is moving from the left side, and the y values will never reach 1. Some numbers would be .9, .99, .999, etc.

This function has a positive sign after the limit 1 of x. This means the limit is moving from the right side, and the y values will also never quite reach 1. Some numbers would be 1.01, 1.001, 1.00001, etc. The graph of a left limitThe graph of a right limit

If the limit of x>3=(3x-9)/(x-3) from the left, lim x->3= (3x-9)/(x-3). The table would look like

x.............|y 2.9........... |.3 2.99........... |.3 from the right.

With the lim x>3+= (3x-9)/(x-3) the table would look like x....................|y 3.1.................|.3 3.11...............|.3

If you continue the table from both sides, you would only get 3. Therefore, the limit of x approaching 3 =3

[|Understanding left and right limits]

Let f be a function of a real variable x. Let c, m be two fixed numbers. If f(x) approaches the values m as x approaches c, we say m is the limit of the function f(x) as x tends to c. This is written as lim x->c f(x) = m

**Left hand limits:**
While defining the limit of a function as x tends to c, we consider values of f(x) when x is very close to c. The values of x may be greater or less than c. If we restrict x to values less than c, then we say that x tends to c from below or from the left and write it symbolically as x ? c ? 0 or simply x ? c?. The limits of f with this restriction on x, is called the left hand limit. This is written as L **f(c) = lim x->c- f(x)**, provided the limit exists.

**Right hand limits:**
Similarly if x takes only values greater than c, then x is said to tend to c from above or from right, and is denoted symbolically as x ? c + 0 or x ? c+.The limits of f is then called the right hand limit. This is written as R **f(c) = lim x-->c+ f(x).** It is important to note that for the existence of lim π>c f(x) it is necessary that both Lf(c) and Rf(c) exists and Lf(c) = Rf(c) =lim x-->c f(x). These left and right hand limits are also known as one sided limits.

Example: when x > 4, | x ? 4 | = x ? 4

Therefore

lim π--->4+ f(x)=lim π->4+ | x? 4 | / x ? 4

=lim π>4+ x ? 4/ x ? 4

Again when x < 4, | x ? 4 | = ? (x ? 4)

Therefore lim π>4- f(x) =lim π>4 ?(x ? 4)/(x ? 4)

=lim π--->4 (? 1) = ? 1

Note that both the left and right hand limits exist but they are not equal.

i.e. Rf(4) =lim π>>4+ f(x)? lim π->4- f(x)= Lf(4).

-ASSESSMENT- [|Limits from Left to Right Assessment]