unit 7

Evaluate the limit by substitution. Sketch a graph of the function. Show that the graph approaches the same value whether you approach x = -1 from the left or the right. lim(x^2-3) x→-1

-2

limits involving approaching infinity: limf(x) x→infinity

important theorem: lim1/x x→infinity =0

0/0 is called an...

indeterminate form because as we approach 0 in the numerator and 0 in the denominator, different things can happen

Derivatives and integrals are...

limits

common trig forms: there are some limits involving trig functions that you should recognize in the future. the most common are:

limsinx/x x→0 =1 limsinax/bx x→0 =a/b lim1-cox/x x→0 =0 limx/cox x→0 =0 *x has to be approaching 0*

Left-hand and right-hand limits are the idea of looking at what happens to a function as you approach a...

particular value of x, from a particular direction. The limit of f(x) as x approaches the value of a from the left is written limf(x) x→a^- and the limit of f(x) as x approaches the value of a from the right is written limf(x) x→a^+

The simplest type of limit involves a...

predictable function (single rule and continuous) in which you can substitute a given x-value and find the corresponding y-value.

approaching, but usually not...

reaching

rationalizing - using the conjugate

recall that the conjugate of a binomical with a radial or radicals can be used to eliminate the radicals. a similar method may be used to evaluate limits of radical expression when substitution does not work

strategies used to evaluate limits

substitution tables and graphs simplifying rational expressions (multiplying/factoring/cancelling) multiplication by a conjugate common trig forms dominant terms

Find the limit if it exists. If the limit doesn't exist, explain why. 1. limsinx x→pi/2 2. limsin4x/x x→0 3. lim(1+3x)/(5-x) x→0 4. lim(1+3x)/(5-x) x→infinity Find each limit by rule or substitution. 5. lim4 x→7 6. lim-25 x→infinity 7. limx^2+2x x→3^+ 8. lim3x-5 x→-1 9. limsin(x) x→pi/2 10. lim2^x x→0^+ 11. limx^2 x→-infinity 12. lim10/x x→infinity 13. lim10/x x→0^+ 14. limsinx/x x→-infinity 15. limsinx/x x→2pi 16. limsinx/x x→0

1. 1 2. 4 3. 0.2 4. -3 5. 4 6. -25 7. 15 8. -8 9. 1 10. 1 11. infinity 12. 0 13. infinity 14. 0 15. 0 16. 1

find: 1. lim(x^2=4x+3)/(x-3) x→3 *use formulas: a^3+b^3 = (a+b)(a^2-ab+b^2) and a^3-b^3 = (a+b)(a^2+ab+b^2) 2. lim(x^2-4)/(x^3+8) x→-2 3. lim (x-3)^2 - 9 / (x) x→0 4. lim(7+h)^2 -49 / (h) h→0

1. 2 2. -1/3 3. -6 4. 14

Find each limit (if it exists). Note: If you cannot find the answer by substitution, try factoring/cancelling and if that doesn't work, take a look at the graph of the function: 1. lim(3x-1) x→100 2. lim(x^2+3x-4) x→-5 3. lim√(x+1) x→80 4. lim(x^2+2x-8)/(x-2) x→-2 5. lim(x^2+2x-8)/(x-2) x→2 6. lim(x-2)/(x^2-4) x→-2 7. lim(√(x+3))/(x+3) x→-3 8. lim(√(x+3))/(x+3) x→97 9. lime^x x→3 10. lim(1/(x-2)) x→0

1. 299 2. 6 3. 9 4. 2 5. 6 6. DNE 7. DNE 8. 1/10 9. 20.09 10. -1/2

evaluate: 1. lim3x/x^+2x x→0 2. lim(x^2+x-6)/(x^2-9) x→3 3. lim(√(x+1)-1)/x x→0 4. lim(√(x+5)-3)/(x-4) x→4 5. limsin3x/x x→0 6. limsin4x/7x x→0 7. lim((h+4)^2 - 16)/h h→0 8. lim1-cos^(2)x/x x→0

1. 3/2 2. DNE 3. 1/2 4. 1/6 5. 3 6. 4/7 7. 8 8. 0

1. lim7+(1/3x)-(2/x^2) x→infinity 2. lim(4x+8)/5x x→infinity 3. lim(3x-1000)/(x+100) x→infinity 4. lim(5x+5)/(7x^2+1) x→infinity 5. lim(5x^2 + 2)/(4x^2 + 7) x→infinity 6. lim(3x^3+5)/(5x^2+1) x→infinity 7. lim(2x^2-4x)/(x+1) x→infinity 8. lim(2x^2-4x)/(x+1) x→-infinity 9. lim(3x^3+2)/(5x^2-1) x→infinity 10. lim(3x^3+2)/(5x^2-1) x→-infinity 11. lim(x^2+2)/(x-555) x→infinity 12. lim(3-2x)/(3x^3-1) x→-infinity 13. lim(3-5x)/(3x-1) x→infinity 14. (3-2x^2)/(3x-1) x→infinity 15. (6x^2-2x-1)/(2x^2+3x+2) x→infinity 16. lim(3x^3+2)/(2x^2-9x^3+7) x→infinity 17. limx/x^2-1 x→-infinity 18. lim(8x^2+3x)/(2x^2-1) x→-infinity 19. lim10-(2/x^2) x→infinity 20. lim4+(3/x) x→-infinity 21. lim(5x^2)/(x+3) x→-infinity 22. lim(1/2x) - (4/x^2) x→infinity 23. limsinx/x x→infinity 24. limcos2x/3x x→infinity

1. 7 2. 4/5 3. 3 4. 0 5. 5-4 6. -infinity 7. infinity 8. -infinity 9. infinity 10. 3/4 11. infinity 12. 0 13. -5/3 14. -infinity 15. 3 16. -1/3 17. 0 18. 4 19. 10 20. 4 21. -infinity 22. infinity 23. 0 24. 0

limits involving infinity (principles of dominance)

1. limx^a/x^b x→infinity, if a<b then, limit = 0 (look for the highest degrees/powers of x) 2. limCx^a/Dx^b x→infinity, if a=b then, limit = C/D (look for the highest degrees/powers of x) 3. limx^a/x^b x→infinity, if a>b then, limit = infinity or -infinity (Look for the highest degrees/powers of x and check the sign of by substituting with a large x-value.)

evaluate: 1. f(x) = {(1/x-1) if x<1; (x^3-2x+5) if x>=1 2. f(x) = {(3-x) if x<2; 2 if x = 2; x/2 if x>2 3. f(x) = {1-x^2 if x<1; 2 if x>=1 4. f(x) = {(2x+1) if x<=-1; 3x if -1<x<1; (2x-1) if x>=1 5. f(x) = {(x-1)^2 if x<0; (x+1)^2 if x>=0

1. not continuous at x = 1 2. not continuous at x = 2 3. not continuous at x = 1 4. not continuous at x = +/-1 5. continuous everywhere

find: lim((√(x+1)-2)/(x-3)) x→3 lim((√(x+8)-2)/(x-4)) x→-4

1/4 1/4

lim(x^2+5x+6)/(x^2+6x+8) x→-1 lim(x^2+5x+6)/(x^2+6x+8) x→-2

2/3 1/2

lim(x^2-4)/(x-2) x→2

4 hole at (2,4)

Evaluate the limit by substitution. Sketch a graph of the function. Show that the graph approaches the same value whether you approach x = 2 from the left or the right. lim(2x+1) x→2

5

substitute: lim((2^x)-1) x→3 lim(2x+1)/(3x-2) x→-2

7 3/8

find: limsin8x/x x→0 limsin2x/3x x→0

8 2/3

What is a Limit?

A limit is a value (the y-value of the function) that a function or sequence "approaches" as the input value (the x-value) approaches some value. Limits are essential to calculus and are used to define continuity, derivatives, and integrals.

When Substitution Doesn't Work : If limits can be found so easily by substitution, then why don't we use substitution all the time? Unfortunately, we can't. Here are 2 functions in which the method of finding limits by substitution doesn't work. lim(x^2+5x+6)/(x^2+6x+8) x→-4 lim(1/(x-2)) x→2

DNE from right: -infinity from left: infinity DNE from right: infinity from left: -infinity

Using Tables to Evaluate Limits

Finding a limit from a table is usually a quick, easy process. When technology is available (and allowed), and the algebra is messy, it's a good idea to pull out your trusty calculator and make a table.

process of continuity:

Process: Step 1: Note where the function could have a discontinuity. This is "c". Step 2: Calculate the right side and left side limits at "c" using the correct notation and compare those limits. If the limits match then the general limit exists. Step 3: Calculate f(c). Step 4: If the general limit and f(c) are the same then the function is continuous at c.

Cancellation before substitution

Sometimes, the substitution that we wish to make is illegal because it makes the denominator 0. Because substituting 2 into the expression would make the denominator 0, we need to factor the numerator and denominator first.

factoring

a hole is referred to as a removable discontinuity. this term indicates that we can use algebra to find the y-value of the hole. be aware that in many calculus problem, the c value that c approaches may give us a hint for factoring

look at page 31-32 for the chocolate problem

cover, do, check, correct

look at page 36 and do

cover, do, check, correct

look at pages 27-29 for using graphs and tables to evaluate limits

cover, do, check, correct

look at pages 45-48 for review problems

cover, do, check, correct

look at pages 43-44 for the tick problem

cover, do, correct, check

continuity is lifting up pen when...

drawing graph