MIDTERM EXAM

Question 1

Let f be a function defined at every number in some open interval containing a, except possibly at the number a itself.

Answer 1

limit

Question 2

if the values of f gets closer and closer to one and only one number Lasxtakes values that are closer and closer to a.

Answer 2

limit

Question 3

The limit of a multiple of a function is simply that multiple of the limit of the function.

Answer 3

constant multiple theorem

Question 4

The limit of a sum of functions is the sum of the limits of the individual functions, and the limit of the difference of functions is the difference of their limits.

Answer 4

addition theorem

Question 5

The limit of a product of functions is equal to the product of the limits.

Answer 5

multiplication theorem

Question 6

The limit of a quotient of functions is equal to the quotient of the limits of the individual functions, provided the denominator limit is not equal to 0.

Answer 6

division theorem

Question 7

The limit of an integer power p of a function is just that power of the limit of the function.

Answer 7

power theorem

Question 8

If n is a positive integer, the limit of the nth root of a function is the nth root of the limit of the function, provided that the nth root of the limit is a real number. If n is even, then L must be positive.

Answer 8

radical or root theorem

Question 9

Let 𝑓 be a function defined on some open interval (𝑎,𝑑). Then the limit of the function 𝑓 as 𝑥 approaches 𝑎 from the right is 𝐿

Answer 9

right hand limit

Question 10

if the values of 𝑓 get closer and closer to one and only one number 𝐿 as 𝑥 approaches 𝑎 through values that are close to but greater than 𝑎.

Answer 10

right hand limit

Question 11

it Let 𝑓 be a function defined on some open interval (𝑐,𝑎). Then the limit of the function 𝑓 as 𝑥 approaches 𝑎 from the left is 𝐿

Answer 11

left hand limit

Question 12

if the values of 𝑓 get closer and closer to one and only one number 𝐿 as 𝑥 approaches 𝑎 through values that are close to but less than 𝑎.

Answer 12

left hand limit

Question 13

as 𝑥 approaches 𝑎, which is written as lim𝑥→𝑎 𝑓(𝑥)=+∞, if the values of 𝑓 get larger and larger as the values of 𝑥 become closer and closer to 𝑎;

Answer 13

𝑓 increases without bound

Question 14

as 𝑥 approaches 𝑎, which is written as lim𝑥→𝑎 𝑓(𝑥)=−∞, if the values of 𝑓 get smaller and smaller as the values of 𝑥 become closer and closer to 𝑎.

Answer 14

𝑓 decreases without bound

Question 15

let f be a function defined on (c,♾️) for some number c. the limit of f as x increases without bound which is L.

if the values of f(x) get closer and closer to one and only one number L as x becomes larger and larger.

Answer 15

limits at infinity

Question 16

A function 𝑓 is said to be _ at 𝑥=𝑎 when all the following conditions are met:
(i) 𝑓(𝑎) is defined
(ii) lim 𝑓(𝑥) exists
(iii) lim 𝑓(𝑥) = 𝑓(𝑎)

Answer 16

continuous

Question 17

This type of discontinuity occurs when function 𝑓 is discontinuous at 𝑎 and condition (ii) is not satisfied.

Answer 17

essential (nonremovable)

Question 18

This type of discontinuity occurs when function 𝑓 satisfies condition (ii) but fails to satisfy either (i) or (iii). Given this type of discontinuity, we can remove the discontinuity by redefining function 𝑓.

Answer 18

removable