Evaluation of Limits MCQ Quiz - Objective Question with Answer for Evaluation of Limits - Download Free PDF

Last updated on Apr 22, 2025

Latest Evaluation of Limits MCQ Objective Questions

Evaluation of Limits Question 1:

If y = In(emx + e-mx), then what is the value of  at x = 0 ?

  1. -1
  2. 0
  3. 1
  4. 2
  5. None of the above

Answer (Detailed Solution Below)

Option 2 : 0

Evaluation of Limits Question 1 Detailed Solution

Concept:

If y = ln f(x) then 
Calculation:

Given function is y = ln(emx + me-mx)

Differentiating, we get

At x = 0, we have

⇒ 

∴ For y = ln(emx + me-mx),  at x = 0.

Evaluation of Limits Question 2:

If  then  is equal to

  1. None of these
  2. None of the above

Answer (Detailed Solution Below)

Option 4 : None of these

Evaluation of Limits Question 2 Detailed Solution

Concept:

Chain Rule: 

  • If y = f(g(x)), then dy/dx = f '(g(x) × g'(x)
  • If y = p(x)/q(x), then dy/dx = 

 

Formula used:

  • (log x)' = 1/x
  • (xn)' = n xn-1

 

Calculation:

If  ,

then, 

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

∴ The correct answer is option (4).

Evaluation of Limits Question 3:

  1. does not exist
  2. exists and it is equal to e3/5
  3. exists and it is equal to e5/3
  4. exists and it is equal to 21/3
  5. None of the above

Answer (Detailed Solution Below)

Option 2 : exists and it is equal to e3/5

Evaluation of Limits Question 3 Detailed Solution

Explanation:

Given:

L = 

Taking log on both side we get

log L =  log(x3 - x2 - 8x + 13)  ----(1)

Taking RHS

Using L'hospital's Rule we get

Again using L'hospital's Rule we get

So we can see that L does not exist.

Evaluation of Limits Question 4:

 is equal to

Answer (Detailed Solution Below)

Option 3 :

Evaluation of Limits Question 4 Detailed Solution

Calculation

Using L’hopital rule 

Hence option 3 is correct

Evaluation of Limits Question 5:

What is the value of ?

  1. 0
  2. 3
  3. 144
  4. 432
  5. 539

Answer (Detailed Solution Below)

Option 4 : 432

Evaluation of Limits Question 5 Detailed Solution

Formula used 

a3 - b= (a - b)(a+ ab + b2)

Calculation

Givn 

x3 - 1728 = (x - 12)(x2 + 12x + 144)

⇒   

⇒  

Now, substituting x = 12 

⇒ 122 + 12 (12) + 144 = 144 + 144 + 144 = 432 

∴ The value of the limit is: 432

Top Evaluation of Limits MCQ Objective Questions

Answer (Detailed Solution Below)

Option 3 : 4

Evaluation of Limits Question 6 Detailed Solution

Download Solution PDF

Concept:

  • 1 - cos 2θ = 2 sin2 θ

 

Calculation:

          (1 - cos 2θ = 2 sin2 θ)

= 4 × 1 = 4

Answer (Detailed Solution Below)

Option 2 : 1

Evaluation of Limits Question 7 Detailed Solution

Download Solution PDF

Concept:

 

Calculation:

As we know  and 

Therefore,  and 

Hence 

Answer (Detailed Solution Below)

Option 3 :

Evaluation of Limits Question 8 Detailed Solution

Download Solution PDF

Calculation:

We have to find the value of 

       [Form ]

This limit is of the form , Here, We can cancel a factor going to ∞  out of the numerator and denominator.

Factor x becomes ∞ at x tends to ∞, So we need to cancel this factor from numerator and denominator.

Answer (Detailed Solution Below)

Option 2 : 1

Evaluation of Limits Question 9 Detailed Solution

Download Solution PDF

Calculation:

We have to find the value of 

       [Form ]

This limit is of the form , Here, We can cancel a factor going to ∞  out of the numerator and denominator.

Factor x2 becomes ∞ at x tends to ∞, So we need to cancel this factor from numerator and denominator.

Answer (Detailed Solution Below)

Option 4 :

Evaluation of Limits Question 10 Detailed Solution

Download Solution PDF

Concept:

  • .
  • .
  • .
  • .

 

Indeterminate Forms: Any expression whose value cannot be defined, like , , 00, ∞0 etc.

  • For the indeterminate form , first try to rationalize by multiplying with the conjugate, or simplify by cancelling some terms in the numerator and denominator. Else, use the L'Hospital's rule.
  • L'Hospital's Rule: For the differentiable functions f(x) and g(x), the , if f(x) and g(x) are both 0 or ±∞ (i.e. an Indeterminate Form) is equal to the  if it exists.

 

Calculation:

 is an indeterminate form. Let us simplify and use the L'Hospital's Rule.

.

We know that , but  is still an indeterminate form, so we use L'Hospital's Rule:

, which is still an indeterminate form, so we use L'Hospital's Rule again:

, which is still an indeterminate form, so we use L'Hospital's Rule again:

.

∴ .

What is  equal to ?

  1. 0
  2. -1
  3. 1
  4. Limit does not exist

Answer (Detailed Solution Below)

Option 1 : 0

Evaluation of Limits Question 11 Detailed Solution

Download Solution PDF

Concept:

log mn = n log m

 

Calculation:

Answer (Detailed Solution Below)

Option 3 : √2

Evaluation of Limits Question 12 Detailed Solution

Download Solution PDF

Formula used:

 

Calculation:

Since, 1 - cos 2θ = sin2θ

⇒ 

 

∴   = √2

Answer (Detailed Solution Below)

Option 3 : π

Evaluation of Limits Question 13 Detailed Solution

Download Solution PDF

Concept:

 

Calculation:

Let 

If x → ∞ then t → 0

= 1 × π 

= π 

Answer (Detailed Solution Below)

Option 1 : -1

Evaluation of Limits Question 14 Detailed Solution

Download Solution PDF

Concept:

 

Calculation:

We have to find the value of 

As we know, 

= 1 ×

=

= -1 × 1

= -1

Answer (Detailed Solution Below)

Option 1 : 3

Evaluation of Limits Question 15 Detailed Solution

Download Solution PDF

This can be written as:

Taking 3n common, we can write:

Here 

So, 

Hot Links: teen patti baaz teen patti download apk teen patti - 3patti cards game