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

Concept:

If y = ln f(x) then 
Calculation:

Given function is y = ln(e^mx + me^-mx)

Differentiating, we get

At x = 0, we have

⇒ 

∴ For y = ln(emx + me-mx),  at x = 0.

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 x^n-1

Calculation:

If  ,

then, 

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

∴ The correct answer is option (4).

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.

Calculation

Using L’hopital rule 

Hence option 3 is correct

Formula used 

a³ - b³ = (a - b)(a² + 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

Concept:

• 1 - cos 2θ = 2 sin2 θ

Calculation:

=           (1 - cos 2θ = 2 sin² θ)

= 

= 

= 4 × 1 = 4

Concept:

Calculation:

As we know  and 

Therefore,  and 

Hence 

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.

= 

= 

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.

= 

= 

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:

.

∴ .

Concept:

log mⁿ = n log m

Calculation:

Formula used:

Calculation:

Since, 1 - cos 2θ = sin²θ

⇒ 

⇒ 

∴   = √2

Concept:

Calculation:

= 

= 

Let 

If x → ∞ then t → 0

= 

= 1 × π 

= π 

Concept:

Calculation:

We have to find the value of 

As we know, 

= 1 ×

=

= 

= -1 × 1

= -1

This can be written as:

Taking 3ⁿ common, we can write:

Here 

So,