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Sample Lecture Notes - University of Massachusetts Amherst

Math131 calculus I The Limit Laws Notes I. The Limit Laws Assumptions: c is a constant and )(limxfax and )(limxgax exist Direct Substitution Property: If f is a polynomial or rational function and a is in the domain of f, then = )(limxfax Simpler Function Property : If )()(xgxf= when ax then)(lim)(limxgxfaxax =, as long as the limit exists. Limit Law in symbols Limit Law in words 1 )(lim)(lim)]()([limxgxfxgxfaxaxax +=+ The limit of a sum is equal to the sum of the limits. 2 )(lim)(lim)]()([limxgxfxgxfaxaxax = The limit of a difference is equal to the difference of the limits.

Math131 Calculus I The Limit Laws Notes 2.3 I. The Limit Laws Assumptions: c is a constant and f x lim ( ) →x a and g x lim ( ) →x a exist Direct Substitution Property: If f is a polynomial or rational function and a is in the domain of f, then = f x lim ( ) x a

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Transcription of Sample Lecture Notes - University of Massachusetts Amherst

1 Math131 calculus I The Limit Laws Notes I. The Limit Laws Assumptions: c is a constant and )(limxfax and )(limxgax exist Direct Substitution Property: If f is a polynomial or rational function and a is in the domain of f, then = )(limxfax Simpler Function Property : If )()(xgxf= when ax then)(lim)(limxgxfaxax =, as long as the limit exists. Limit Law in symbols Limit Law in words 1 )(lim)(lim)]()([limxgxfxgxfaxaxax +=+ The limit of a sum is equal to the sum of the limits. 2 )(lim)(lim)]()([limxgxfxgxfaxaxax = The limit of a difference is equal to the difference of the limits.

2 3 )(lim)(limxfcxcfaxax = The limit of a constant times a function is equal to the constant times the limit of the function. 4 )](lim)(lim)]()([limxgxfxgxfaxaxax = The limit of a product is equal to the product of the limits. 5 )(lim)(lim)()(limxgxfxgxfaxaxax = ()0)(lim xgifax The limit of a quotient is equal to the quotient of the limits. 6 naxnaxxfxf)](lim[)]([lim = where n is a positive integer 7 ccax= lim The limit of a constant function is equal to the constant. 8 axax= lim The limit of a linear function is equal to the number x is approaching. 9 nnaxax= lim where n is a positive integer 10 nnaxax= lim where n is a positive integer & if n is even, we assume that a > 0 11 naxnaxxfxf)(lim)(lim = where n is a positive integer & if n is even, we assume that )(limxfax > 0 Math131 calculus I Notes page 2 ex#1 Given2)(lim3= xfx,1)(lim3 = xgx, 3)(lim3= xhx use the Limit Laws find )()()(lim23xgxxhxfx ex#2 Evaluate 4612lim222 ++ xxxx, if it exists, by using the Limit Laws.

3 Ex#3 Evaluate: 532lim21 + xxx ex#4 Evaluate: xxx20)1(1lim ex#5 Evaluate: hhh24lim0 + Math131 calculus I Notes page 3 Two Interesting Functions 1. Absolute Value Function Definition: < =0 if 0 if xxxxx 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?

4 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~2. The Greatest Integer Function Definition: [][]=x the largest integer that is less than or equal to x. ex 6 [[ 5 ]] = ex 7 [[ ]] = ex 8 [[ 3 ]] = ex 9 [[ ]] = ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Theorem 1: Lxfax= )(lim if and only if )(lim)(limxfLxfaxax+ == ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ex#10 Prove that the xxx0lim does not exist. Math131 calculus I Notes page 4 ex#11 What is ]][[lim3xx ?

5 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 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 )(lim)(limxgxfaxax . ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ex12 Find xxx1sinlim20 . To find this limit, let s start by graphing it. Use your graphing calculator. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The Squeeze Theorem: If )()()(xhxgxf when x is near a (except possibly at a) and Lxhxfaxax== )(lim)(lim then Lxgax= )(lim Math131 calculus I Limits at Infinity & Horizontal Asymptotes Notes Definitions of Limits at Large Numbers Theorem If r > 0 is a rational number then 01lim= rxx If r > 0 is a rational number such that rxis defined for all x then 01lim= rxx Definition in Words Precise Mathematical Definition Large POSITIVE numbers Let f be a function defined on some interval (a, ).

6 Then Lxfx= )(lim means that the values of f(x) can be made arbitrarily close to L by taking x sufficiently large in a positive direction. Let f be a function defined on some interval (a, ). Then Lxfx= )(lim if for every > 0 there is a corresponding number N such that if x > N then < Lxf)( Large NEGATIVE numbers Let f be a function defined on some interval (- ,a). ). Then Lxfx= )(lim means that the values of f(x) can be made arbitrarily close to L by taking x sufficiently large in a negative direction. Let f be a function defined on some interval (- ,a).

7 Then Lxfx= )(lim if for every > 0 there is a corresponding number N such that if x < N then < Lxf)( Definition What this can look Horizontal Asymptote The line y = L is a horizontal asymptote of the curve y = f(x) if either is true: 1. Lxfx= )(lim or 2. Lxfx= )(lim Vertical Asymptote The line x = a is a vertical asymptote of the curve y = f(x) if at least one of the following is true: 1. = )(limxfax 2. = )(limxfax 3. =+ )(limxfax 4. = )(limxfax 5. = )(limxfax 6. =+ )(limxfax Math131 calculus I Notes page 2 ex#1 Find the limit: 53limxx ex#2 Find the limit: 452lim233+ + xxxxx ex#3 Find the limit: xxxx39lim2 + ex#4 Find the limit: xxcoslim Math131 calculus I Notes page 3 ex#5 Find the vertical and horizontal asymptotes of the graph of the function:5312)(2 +=xxxf.

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