Finding Limits

Question 1

How to say x -> 1

Answer 1

x approaches 1

Question 2

How to say x -> 1-?

Answer 2

x approaches 1 from the left

Question 3

How to say x -> 1+?

Answer 3

limit as x approaches 1 from the right

Question 4

0/0 description or categorization

Answer 4

Indeterminate Form

Question 5

How to use the table method to numerically determine a limit?

Answer 5

First, write out a table centered around the value the limits will be approaching.

Then, fill the table with the resulting values on each side. If the numbers you chose are getting arbitrarily small as they approach the number in question they should reveal what the limit is.

Question 6

Notation for limit as x approaches 1 equals 3

Answer 6

Question 7

Causes of limits not existing (DNE)

Answer 7

1. Approaching different values
   
   1. Given value B, the limit as x approaches B from the left does not equal the limit as x approaches B from the right
2. Oscillating Behavior
   
   1. Given value B, the limit as x approaches B from the left is impossible to determine because of oscillating behavior OR the limit as x approaches B from the right is impossible to determine because of oscillating behavior
   2. It is impossible to determine because it is not consistently approaching a value.
      
      1. Example: limit as x approaches 0 of sin(1/x)

Question 8

How to use the graph method.

Answer 8

Given a graph of function f(x), it can be fairly easy to determine the limit of a given y value. Graphs cannot always be trusted as they have a difficult time representing oscillations and holes, but for functions that don’t encounter these situations, they can be an excellent way to quickly determine limits.

Question 9

What are the three basic properties of limits?

Answer 9

Given that b and c are real numbers and n is a positive integer, the following is true of limits:

1. The limit as x approaches c (a real number) when f(x) is b (a real number) is b itself.
   
   1. This equation has no slope, it is a horizontal line. No matter what x is approaching, b is not changing.
2. The limit as x approaches c (a real number) when f(x) is x is c.
   
   1. This is a 45 degree line passing through the origin (y=x)
3. The limit as x approaches c (a real number) when f(x) = x² is c²

Question 10

What are the intermediate properties of limits?

Answer 10

1. The limit of a function multiplied by a real number b is equal to the limit of the function, multiplied by b. Essentially, multiplying the function by some number and finding that limit will be equivalent to finding the limit of the function and then multiplying the limit by the number.
2. The limit of two functions being added or subtracted from one another is the same as adding or subtracting the limits of each function.
3. The limit of two functions being multiplied is equivalent to the limit of each function being multiplied.
4. The limit of one function dividing another is equivalent to the limits of each function divided by one another (as long as the numerator and denominator don’t swap)
5. The limit of a function raised to an integer is the same as raising the limit of a function to an integer.

Question 11

Direct Substitution

Answer 11

If there are no domain issues for the x value we are approaching, we can attempt to plug in the x value to the function directly and this will give us the limit itself.

More complex definition: Given that f(x) is a polynomial function and c is a real # then the limit as x approaches c is f( c).

This holds true for rational functions as well (division of variables) except when the denominator is 0!

Question 12

0/0 vs x/0 where x is not 0

Answer 12

0/0 is an indeterminate form

x/0 is undefined, which means it is a domain issue.

Question 13

Theorem for direct substitution with radicals?

Answer 13

Given that n is a positive integer, direct substitution works for radicals for all possible values if n is odd and for values greater than 0 when n is even.

Question 14

Limit as x approaches 0 of the square root of x?

Answer 14

Does not exist! It may seem like it is 0 but it is not. It exists from the right but not from the left.

Question 15

Direct substitution with trig functions

Answer 15

Direct substitution works for functions that use trig functions as long as the value x is approaching in the limit is:

1. a real #
   
   1. existent in the domain of the trig function

Question 16

Explain how factor and cancel methods apply to limits

Answer 16

Factor and cancel of rational functions allows us to identify at which point a function does not exist. In precalculus this would allow us to identify the point, in calculus we can use the reduced function to find the limit because the limit is not the actual value. For instance, while the function has no value at 4, the function usually tells us what it is doing just before and after 4.

Question 17

Explain the rationilizing technique

Answer 17

We can multiply the numerator and denominator of a function by it’s conjugal when one has a radical, with partial factorization this can often reveal an opportunity for cancellation which enables us to use the factor and cancel reduction method of determining a limit.

Question 18

What is the squeeze (sandwich) theorem?

Answer 18

If h(x) ≤ f(x) ≤ g(x) for all x in an open interval containing a real number c, except maybe at c itself (f(c ) could be undefined) and if the limit of h(x) as x approaches c equals L which also equals the limit of g(x) as x approaches c, then the limit of f(x) as x approaches c is L.

It essentially gets squeezed between the other two functions given the greater than less than are true.

Question 19

Limit as x approaches 0 of sinx/x

Answer 19

1

Question 20

Limit as x approaches 0 of (1-cos(x)/x)

Answer 20

0

Question 21

Strategic approach to finding limits

Answer 21

1. Try direct substitution
2. Rational Functions
   
   1. Factor and cancel
3. Radical Functions
   
   1. Rationalize
4. Trig functions
   
   1. hunt for squeeze theorem limits
5. Sketch a graph (slow)
6. Make a table (slowest)