Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:

Calculating Limits Finding the limit of a function f a point x = a. Distinguishing the following cases: The case when f is continuous a x = a. The case 0/0. The case ∞/ ∞ The case of an infinite limit The case c/∞, where c is a real number. The case, when it is possible to use the squeeze theorem.

1. The case when f is continuous at x = a If f is continues at x=a, then: Notice: 1. Polynomial functions and the cubic root function ( & all functions of its two families) are everywhere continuous. 2. Rational, trigonometric and root functions are continuous at every point of their domains. 3. If f and g are continuous a x=a, then so are cf, f+g, f-g, fg and f/g (provided that he limit of f at x=a is not zero)

Examples for the case when f is continuous at x = a

Examples for the case when f is continuous at x = a

Examples for the case when f is continuous at x= a

2. The case 0/0 Suppose we want to find: For the case when: Then this is called the case 0/0. Caution: The limit is not equal 0/0. This is just a name that classifies the type of limits having such property.

Examples for the case 0/0

Examples for the case 0/0

Examples for the case 0/0

Question: Simplify the formula of f and graph it!

Questions

3. The case ∞/ ∞ Suppose we want to find: For the case when the limits of both functions f and g are infinite Then this is called the case ∞/ ∞. Caution: The limit is not equal ∞/∞. This is just a name that classifies the type of limits having such property.

Limits at infinity A function y=f(x) may approach a real number b as x increases or decreases with no bound. When this happens, we say that f has a limit at infinity, and that the line y=b is a horizontal asymptote for f.

Limit at infinity: The Case of Rational Functions A rational function r(x) = p(x)/q(x) has a limit at infinity if the degree of p(x) is equal or less than the degree of q(x). A rational function r(x) = p(x)/q(x) does not have a limit at infinity (but has rather infinite right and left limits) if the degree of p(x) is greater than the degree of q(x).

Example (1) Let Find Solution: Since the degree of the polynomial in the numerator, which is 9, is equal to the degree of the polynomial in the denominator, then

To show that, we follow the following steps:

Example (2) Let Find Solution: Since the degree of the polynomial in the numerator, which is 9, is less than the degree of the polynomial in the denominator, which is 12, then

To show that, we follow the following steps:

Example (3) Let Find Solution: Since the degree of the polynomial in the numerator, which is 12, is greater than the degree of the polynomial in the denominator, which is 9, then

They are infinite limits. To show that, we follow the following steps:

Example (4) Let Find Solution: Since the degree of the polynomial in the numerator, which is 12, is greater than the degree of the polynomial in the denominator, which is 9, then

They are infinite limits. To show that, we follow the following steps:

Example (5) Let Find Solution: Since the degree of the polynomial in the numerator, which is 12, is greater than the degree of the polynomial in the denominator, which is 8, then

They are infinite limits. To show that, we follow the following steps:

Example (6) Let Find Solution: Since the degree of the polynomial in the numerator, which is 12, is greater than the degree of the polynomial in the denominator, which is 8, then

They are infinite limits. To show that, we follow the following steps:

Limits & Infinity Problems Involving Roots

Introduction We know that: √x2 = |x| , which is equal x is x non-negative and equal to – x if x is negative For if x = 2, then √(2)2 = √4 = 2 = |2| & if x = - 2, then √(-2)2 = √4 = -(-2) =|-2|

Example

Example

4. The case of infinite limit

Infinite Limits A function f may increases or decreases with no bound near certain values c for the independent variable x. When this happens, we say that f has an infinite limit, and that f has a vertical asymptote at x = c The line x=c is called a vertical asymptote for f.

Infinite Limits A function has an infinite one-sided limit at a point x=c if at that point the considered one-sided limit of the denominator is zero and that of the numerator is not zero. The sign of the infinite limit is determined by the sign of both the numerator and the denominator at values close to the considered point x=c approached by the variable x (from the considered side).

Infinite Limits- The Case of Rational Functions A rational function has an infinite one-sided limit at a point x=c if c a zero of the denominator but not of the numerator. The sign of the infinite limit is determined by the sign of both the numerator and the denominator at values close to the considered point x=c approached by the variable x (from the considered side).

Example (1) Let Find Solution: First x=0 is a zero of the denominator which is not a zero of the numerator.

a. As x approaches 0 from the right, the numerator is always positive ( it is equal to 1) and the denominator approaches 0 while keeping positive; hence, the function increases with no bound. Thus: The function has a vertical asymptote at x = 0, which is the line x = 0 (see the graph in the file on basic algebraic functions). b. As x approaches 0 from the left, the numerator is always positive ( it is equal to 1) and the denominator approaches 0 while keeping negative; hence, the function decreases with no bound.

Example (2) Let Find Solution: First x=1 is a zero of the denominator which is not a zero of the numerator.

a. As x approaches 1 from the right, the numerator approaches 6 (thus keeping positive), and the denominator approaches 0 while keeping positive; hence, the function increases with no bound. Thus, The function has a vertical asymptote at x = 1, which is the line x = 1 b. As x approaches 1 from the left, the numerator approaches 6 (thus keeping positive), and the denominator approaches 0 while keeping negative; hence, the function decreases with no bound. The function has a vertical asymptote at x=1, which is the line x = 1. Thus:

Example (3) Let Find Solution: First x=3 is a zero of the denominator which is not a zero of the numerator.

6. The case constant/∞ Suppose we want to find: For the case when: In this case, no mater what the formulas of g and h are, we will always have: Then this is called the case c/∞. Caution: The limit is not equal c/ ∞. This is just a name that classifies the type of limits having such property. This limit is always equal zero

Example on the case constant/∞

6. Using the Squeeze Theorem

The Squeeze (Sandwich or Pinching)) Theorem Suppose that we want to find the limit of a function f at a given point x=a and that the values of f on some interval containing this point (with the possible exception of that point) lie between the values of a couple of functions g and h whose limits at x=a are equal. The squeeze theorem says that in this case the limit of f at x=a will equal the limit of g and h at this point.

The Squeeze Theory

Example (1)

Example (2)

Example (3)

Example (4)

Example (5)

Example (6)

Example (7)

Example (8)

Example (9)