An introduction to limits Limits in calculus : This section gives some examples of how to use algebraic techniques to compute limits. these In cludethe terms of an infinite series, the sum of an infinite series,the limit of afunction, The slope of aline tangent to the graph of afunction, the area of aregion bounded by the graphs of several functions.

Find the sum of the terms of the series mentioned above as n gets very large the sum of the series is so limt

Try some values for x close to 2 From the chart it appears that as x tends to 2 F(x) tends to zero so

compute the slopes of the secant lines

So as ∆ x tend to 2 ero so the slope of the line tangent at x=3 close and close to 2 for any point p close to T the slope PT is given by Or x tends to zero p close to T so the slope of line tangent at point T

The expression in the left side is known as the derivative of f (x) at x = 3, and is denoted by f '(3). and we will learn many techniques for the derivative in the following chapters In the following chapter Area under a curve :-

Then the slope of the line tangent to the graph of f(x) = (x-2) at x = 3 is 2 The area below the curve F(x) = x 2, above the x ax is right the line x=1 and left the line x=5

Using four inscribed rectangles, each having a base of 1 unit, their corresponding heights are found: f(1) = 1, f (2) = 4, f (3) = 9, and f (4) = 16. The area computation is shown at the right.... area = 1*1 +1*4+1*9+1*16 =30 This area is less than the actual desired area.

Use four rectangular each having abase of 1 unit Their corresponding height are found : f(2)=4, f(3)=9, f(4)=16 and f(5)=25... area = 1*4+1*9+1*16+1*25=54 This area is greater than the actual desired area. So the actual area is greater than 30 and less than 54

The actual area of the region described in Step 1 is greater than 30 and less than 54. If you wanted a closer approximation of the actual area, you would use a very large number of rectangles, each having a base of, where n is the number of rectangles used. The corresponding height for each rectangle would then be f(xi), where i represents the 1st, or 2nd, or 3rd, or 4th rectangle of the n rectangles used. The sum of the areas of these rectangles is represented by

The actual area would be found by letting n →, so and then find the limt → 0

you will compute these sorts of areas, after learning some techniques of integration.

DEFINITION OF THE LIMIT OF A FUNCTION let f(x) approaches L as x tends to c if Whereand These means that the distance from f(x) to l can be made small by making the distance from x to c small but not 0. UsingDefinition to verify alimit using f(x) Use the definition to verify that whenever you need to find a connection between

If you move left and right of x = 3 just 1 unit, x would be in the interval (2,4) so that

One sided limits :-

EX :- Note: When you write IT does not mean that the limit exists. The limit actually does not exist because f (x) increases without bound as x approaches c.

ONE-SIDED LIMITS FOR A CONDITIONAL FUNCTION From the left, you can see that And From the right, you can see that

ONE-SIDED LIMITS FOR A POLYNOMIAL FUNCTION

Determine Limits from the Graph of a Function As x approaches -1 from the left and then from the right, two different limits so is non exist.

From left at right

LIMITS OF A TRIGONOMETRIC FUNCTION To the right is the graph of f(x) = sinx. Determine each limit below. lim (sinx) is nonexistent, sinx oscillates between –1 and 1.

Properties of Limits :- Let k and c be constants, let n be a positive integer, and let f and g be functions such that Ex :/

2

3

Special Trigonometric Limits: Ex :/

Definition of a Function Continuous at a Point The function f is continuous at the number c if the following conditions are satisfied: Another test for continuity by viewing the function’s graph is that the graph has no holes, no jumps, and no vertical asymptotes. An example of the first figure is

Ex :/

Condition iii of the definition fails.Note the “hole” at x = 3. This is also known as a removable discontinuity.