Test: Introduction And Algebra Of Limits

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QUESTION: 1

If the right and left hand limits coincide, we call that common value as the limit of f(x) at x = a and denote it by

Solution:

If the right-hand and left-hand limits coincide, we say the common value as the limit of f(x) at x = a and denote it by limx→a f(x) = l

If limx→a- f(x) is the expected value of f at x = a given the values of ‘f’ near x to the left of a. This value is known as the left-hand limit of ‘f’ at a.
If limx→a+ f(x) is the expected value of f at x = a given the values of ‘f’ near x to the right of a. This value is known as the right-hand limit of f(x) at a.

QUESTION: 2

If f(x) = 2, then

A.
2
B.
1
C.
4
D.
0

Solution:

f(x) = 2
Hence it doesnt contain any variable
so, lim(x → 2) f(x) = 2

QUESTION: 3

A.
π – 22/7
B.
π
C.
0
D.
(7x-22)/7

Solution:

lim(x → π ) (x - 22/7)
Taking limit, we get
⇒ π - 22/7

QUESTION: 4

For the limit of a function to exist we must have

A.
LHL = RHL
B.
LHL = 2 RHL
C.
LHL < RHL
D.
LHL > RHL

Solution:

Recall for a limit to exist, the left and right limits must exist (be finite) and be equal.

QUESTION: 5

Let f be any function, such that exists, then

A.
λℓ
B.
ℓ
C.
1
D.
λ

Solution:

lim(x → a) f(x) = l
λf(x) = λl

QUESTION: 6

Let f(x) and g(x) be two function, such that and exists, then the limit of the product of the function f(x) and g(x) is given by

Solution:

By algebric property of limits,
Limit of products = product of limits
lim(x → a) (f(x)*g(x)) = lim(x → a) f(x) * lim(x → a) g(x)

QUESTION: 7

is the expected value of f at x = a given the values of f near x to the left of a. This value is called the……….of f at a.

A.
Limit
B.
Left Hand limit
C.
Right Hand Limit
D.
Value of function

Solution:

QUESTION: 8

As x → a, f(x) → l, then l is called the……..of the function f(X) which is symbolically written as…….

Solution:

The number L is called the limit of function f(x) as x → a if and only if, for every ε>0 there exists δ>0
which is written as
lim (x → a) |f(x) − l|
lim (x → a) f(x) = l

QUESTION: 9

The value of

A.
3/5
B.
3/2
C.
3/4
D.
2/5

Solution:

QUESTION: 10

Consider the function f(x) = x + 10. Let us compute the value of the function f(x) for x very near to 5. Some of the points near and to the left of 5 and right to the 5 are given in the table.