Math131 Calculus I The Limit Laws Notes 2.3

I. The Limit Laws Assumptions: c is a constant and xf )(lim and xg )(lim exist →ax →ax

Limit Law in symbols Limit Law in words The limit of a sum is equal to 1 +=+ xgxfxgxf )(lim)(lim)]()([lim →ax →ax →ax the sum of the limits. The limit of a difference is equal to 2 −=− xgxfxgxf )(lim)(lim)]()([lim →ax →ax →ax the difference of the limits. The limit of a constant times a function is equal 3 = xfcxcf )(lim)(lim →ax →ax to the constant times the limit of the function. The limit of a product is equal to 4 ⋅= xgxfxgxf )](lim)(lim)]()([lim →ax →ax →ax the product of the limits. xf )(lim xf )( →ax The limit of a quotient is equal to 5 lim = ( xgif ≠ 0)(lim ) →ax xg )( xg )(lim →ax the quotient of the limits. →ax

n n 6 = xfxf )](lim[)]([lim where n is a positive integer →ax →ax

The limit of a constant function is equal 7 lim = cc →ax to the constant. The limit of a linear function is equal 8 lim = ax →ax to the number x is approaching.

nn 9 lim = ax where n is a positive integer →ax

nn where n is a positive integer & if n is even, 10 lim = ax →ax we assume that a > 0 where n is a positive integer & if n is even, 11 n = n xfxf )(lim)(lim →ax →ax we assume that xf )(lim > 0 →ax

Direct Substitution Property: If f is a polynomial or rational function and a is in the domain of f, then xf )(lim = →ax

“Simpler Function Property”: If = xgxf )()( when x ≠ a then = xgxf )(lim)(lim , as long as the →ax →ax limit exists. Math131 Calculus I Notes 2.3 page 2 ex#1 Given xf = 2)(lim , xg −= 1)(lim , xh = 3)(lim use the Limit Laws find − 2 xgxxhxf )()()(lim x→3 x→3 x→3 x→3

x 2 +12 ex#2 Evaluate lim , if it exists, by using the Limit Laws. x→2 x 2 + 6x − 4

ex#3 Evaluate: 2 xx −+ 532lim x→1

−− x)1(1 2 ex#4 Evaluate: lim x→0 x

h −+ 24 ex#5 Evaluate: lim h→0 h Math131 Calculus I Notes 2.3 page 3

Two Interesting Functions

1. Absolute Value Function

 if xx ≥ 0 Definition: x =   if xx <− 0

Geometrically: The absolute value of a number indicates its distance from another number.

=− acx means the number x is exactly _____ units away from the number _____.

<− acx means: The number x is within _____ units of the number _____.

How to solve equations and inequalities involving absolute value: Solve: |3x + 2| = 7 Solve: |x - 5| < 2

What does |x - 5| < 2 mean geometrically?

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 2. The Greatest Integer Function Definition: [[x]] = the largest integer that is less than or equal to x.

ex 6 [[ 5 ]] = ex 7 [[ 5.999 ]] = ex 8 [[ 3 ]] = ex 9 [[ -2.6 ]] =

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Theorem 1: )(lim = Lxf if and only if = = xfLxf )(lim)(lim →ax →ax − ax +→ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ x ex#10 Prove that the lim does not exist. x→0 x Math131 Calculus I Notes 2.3 page 4 ex#11 What is x]][[lim ? x→ 3

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Theorem 2: If ≤ xgxf )()( when x is near a (except possibly at a) and the limits of f and g both exist as x approaches a then ≤ xgxf )(lim)(lim . →ax →ax

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 1 ex12 Find x2 sinlim . To find this limit, let’s start by graphing it. Use your graphing calculator. x→0 x

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The Squeeze Theorem: If ≤ ≤ xhxgxf )()()( when x is near a (except possibly at a) and = )(lim)(lim = Lxhxf then )(lim = Lxg →ax →ax →ax

Math131 Calculus I Limits at Infinity & Horizontal Asymptotes Notes 2.6

Definitions of Limits at Large Numbers

Definition in Words Precise Mathematical Definition Let f be a function defined on some interval (a, ∞).

Let f be a function defined on some interval (a, ∞). Then )(lim = Lxf if for every ε > 0 there is a Then )(lim = Lxf means that the values of f(x) can x ∞→ x ∞→

Large be made arbitrarily close to L by taking x sufficiently corresponding number N such that if x > N then numbers POSITIVE large in a positive direction. )( Lxf <− ε Let f be a function defined on some interval Let f be a function defined on some interval (-∞,a). Then )(lim = Lxf if for every ε > 0 there (-∞,a). ∞). Then )(lim = Lxf means that the x −∞→ x −∞→

Large values of f(x) can be made arbitrarily close to L by is a corresponding number N such that if x < N then numbers NEGATIVE taking x sufficiently large in a negative direction. )( Lxf <− ε

Definition What this can look like…

The line y = L is a horizontal asymptote of the curve y = f(x) if either is true: 1. )(lim = Lxf x ∞→ or 2. )(lim= Lxf

Horizontal Asymptote x −∞→

The line x = a is a vertical asymptote of the curve y = f(x) if at least one of the following is true: 1. xf )(lim ∞= →ax 2. xf )(lim ∞= →ax − 3. xf )(lim ∞= →ax +

Vertical AsymptoteVertical 4. xf )(lim −∞= →ax 5. xf )(lim −∞= →ax − 6. xf )(lim −∞= →ax +

Theorem 1 • If r > 0 is a rational number then lim = 0 x ∞→ x r 1 • If r > 0 is a rational number such that x r is defined for all x then lim = 0 x −∞→ x r

Math131 Calculus I Notes 2.6 page 2

3 ex#1 Find the limit: lim x ∞→ x 5

3 + 2 xx ex#2 Find the limit: lim x ∞→ 5 x x 23 +− 4

ex#3 Find the limit: 2 −+ 39lim xxx x ∞→

ex#4 Find the limit: coslim x x ∞→ Math131 Calculus I Notes 2.6 page 3

x 2 +12 ex#5 Find the vertical and horizontal asymptotes of the graph of the function: xf )( = 3x − 5

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