Calculus and Analytical Geometry Lecture # 5 MTH 104

Limits The concept of limit is the fundamental building block on which all calculus concepts are based. We shall develop concept of limits both numerically and graphically and then techniques of evaluating limits will be discussed. Tangent line The line through P and Q is called a secant line for the circle at P. As Q moves along the curve towards P, then the secant line will rotate toward a limiting position. The line in this limiting position is called Tangent line.

Example Find an equation for the tangent line to the parabola at the point P(1,1). solution Consider the secant line through P(1,1) and on the parabola that is distinct from P. The slope of this secant line is Q gets closer and closer to P means x gets closer and closer to 1

The slope of secant line gets closer and closer to 2 as x gets closer and closer to 1 Hence the equation if tangent line is

Limits---Numerically The most basic use of limits is to describe how a function behaves as the independent variable approaches a given variable. Consider Examine the behavior of f (x) when the value of x gets closer and closer to 2? x f(x) Approaching to 2 from the left side

Limits---Numerically x f(x) Approaching to 2 from the right side We write

Limits---Graphically

limits An informal view: If the values of f(x) can be made as close as we like to L by taking values of x sufficiently close to a ( but not equal to a), then we write Example: x f(x) Left side Right side

One sided limits

An informal view: If the values of f(x) can be made as close as we like to L by taking values of x sufficiently close to a ( but greater than a), then we write And if the values of f(x) can be made as close as we like to L by taking values of x sufficiently close to a ( but less than a), then we write

Relationship between one-sided and two sided limits The two sided limit of a function f(x) exists at a if and only if both of the one sided limits exists at a and have the value; that is

Infinite limits Sometimes one sided or two sided limits fail to exist because the values of the function increases or decreases without bound.

Infinite limits

An informal view:

Vertical asymptotes The line x = a is called a vertical asymptote of the curve y =f(x) if at least one of the following statements is true

Vertical asymptotes

Example

Computing limits We will use limit laws to calculate limits The limit laws

The Squeezing Theorem

Example: show that

Computing limits