2.1 Limits

Limit of a Continuous Function

If f with hook of x is a continuous function for all real numbers, then lim over x right arrow c of f with hook of x equals f with hook of x times c

Limits of Rational Functions

A. If f with hook of x is a rational function given by f with hook of x equals p of x over q of x, such that p of x and q of x have no common factors, and c is a real number such that q of c = 0, then

I. lim over x right arrow c of f with hook of x does not exist

II. lim over x right arrow c of f with hook of x = plus or minus infinity right arrow x equals c is a vertical asymptote

B. If f with hook of x is a rational function given by f with hook of x equals p of x over q of x comma such that reducing a common factor between p of x and q of x results in the agreeable function k of x, then

I. lim over x right arrow c of f with hook of x = lim over x right arrow c of p of x over q of x equals lim over x right arrow c of k of x = k of c right arrow Hole at the point open paren c comma k of c close paren

Limits of a Function as x Approaches Infinity

If f with hook of x is a rational function given by (x) = p of x over q of x, such that p of x and q of x are both polynomial functions, then

A. If the degree of p of x > q of x, lim over x right arrow infinity of f with hook of x equals infinity

B. If the degree of p of x is less than q of x, lim over x right arrow infinity of f with hook of x equals infinity right arrow y equals 0 is a horizontal asymptote

C. If the degree of p of x equals q of x, lim over x right arrow infinity of f with hook of x equals c, where c is the ratio of the leading coefficients right arrow y equals c is a horizontal asymptote

Special Trig Limits

L'Hospital's Rule

If results lim over x right arrow c of f with hook of x or lim over x right arrow infinity of f with hook of x results in an indeterminate form open paren 0 over 0 comma the fraction with numerator infinity and denominator infinity comma infinity minus infinity comma 0 times infinity comma 1 raised to the infinity power comma infinity to the 0 power close paren, and f with hook of x equals p of x over q of x, right double arrow

lim over x right arrow c of f with hook of x equals lim over x right arrow c of p of x over q of x equals lim over x right arrow c of p prime of x over q prime of x and lim over x right arrow infinity of f with hook of x equals lim over x right arrow infinity of p of x over q of x equals lim over x right arrow infinity of p prime of x over q prime of x

The Definition of Continuity

A function f with hook of x is continuous at c if:

I. lim over x right arrow c of f with hook of x exists

II. f with hook of c exists

III. lim over x right arrow c of f with hook of x equals f with hook of c

Types of Discontinuities

Removable Discontinuities (Holes):

I. lim over x right arrow c of f with hook of x equals L (the limit exists)

II. f with hook of c is undefined

Non-Removable Discontinuities (Jumps and Asymptotes):

A. Jumps:

lim over x right arrow c of f with hook of x = DNE because lim over x right arrow c raised to the minus power of f with hook of x is not equal to lim over x right arrow c raised to the plus power of f with hook of x

B. Asymptotes (Infinite DIscontinuities):

lim over x right arrow c of f with hook of x equals plus or minus infinity

Intermediate Value Theorem

If f with hook is a continuous function on the closed interval open bracket a comma b close bracket and k is any number between f with hook of a and f with hook of b, then there exists at least one value of c on open bracket a comma b close bracket such that f with hook of C equals k . In other words, on a continuous function, if f with hook of a is less than f with hook of b, any y value greater than f with hook of a and less than f with hook of b is guaranteed to exist on the function f with hook.

Average Rate of Change

The average rate of change, m, of a function f with hook on the interval open bracket a comma b close bracket is given by the slope of the secant line.

m equals the fraction with numerator f with hook of b minus f with hook of a and denominator b minus a

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