Limit of functions

LIMIT OF FUNCTIONS

One-sided limit, limit, existence of limit, limit at infinity, infinite limit

One-sided limit

Given a function defined by

Is the value y (real number) being approached by f(x) as x gets closer and closer to a from the right.

One-sided limit

Given a function defined by

Is the value y (real number) “being approached” by f(x) as x gets closer and closer to a from the left.

One-sided limit

Consider the function defined by

a. Evaluate b. Evaluate

One-sided limit

03 03

One-sided limit

392

x

xxf

x 392

x

xxf x 392

x

xxf

-3.1 -6.1 -2.9 -5.9 -3.01 -6.01 -2.99 -5.99 -3.001 -6.001 -2.999 -5.999 -3.0001 -6.0001 -2.9999 -5.9999

Table 2.1 Some numerical computations close to -3 from the left and right

One-sided limit

392

x

xxf

Theorem 2.1 Existence of LimitThe limit of a function exists if and only if the one-sided limits of the function are equal

One-sided limit

Given a function defined by

One-sided limit

Functions whose limit at a does not exist.Example 2.1

Evaluate , does it exist?

One-sided limit

a. Evaluate .b. Are they equal?

Conclusion: the limit does not exist

One-sided limit

Function whose limit does not exist at a.Example 2.4Evaluate where

One-sided limit

The limits do not point to a specific real number. Conclusion: limit does not exist as x approaches 3

Limit Theorems

Limit at infinity

Evaluate and , where

Limit at infinity

As x approaches positive infinity or negative infinity, the quotient approaches zero.

Limit at infinity

Evaluate where

Solution

Limit at Infinity and Horizontal AsymptoteObtain the horizontal asymptote of

Solution

Limit at infinity and horizontal asymptote

Horizontal Asymptote

Limit at infinity and asymptotes (horizontal and oblique)

Limit at infinity and asymptotes (horizontal and oblique)

Examples

Limit at infinity and asymptotes (horizontal and oblique) Perform long division

on

Limit of some trigonometric functions

Theorems

Limit of trigonometric functions

Limit of trigonometric functions

Limit of trigonometric functions

Limit of trigonometric functions