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

Calculation:

Given,

The function is .

Statement I: f(x) is an increasing function.

The derivative of f(x) is:

When 0 )\), 0 )\), so (f(x) is increasing.

When (x

At (x = 0), ( f'(x) = 0 ), meaning the function is neither increasing nor decreasing at this point.

Hence, f(x)  is not entirely increasing. It is increasing for (x > 0) and decreasing for ( x

Statement II: f(x) has local maximum at x = 0

Since the function is a parabola opening upwards (because the coefficient of x² is positive), it has a global minimum at x = 0, not a local maximum.

Conclusion:

- Statement I is incorrect because the function is not entirely increasing. It is increasing for x > 0  and decreasing for  x

- Statement II is incorrect because the function has a global minimum at x = 0, not a local maximum.

Hence, the correct answer is Option 4. 

Calculation:

Given,

The function is and .

We are tasked with finding:

Multiply both the numerator and denominator by their respective conjugates:

Simplify the numerator:

Simplify the denominator:

Now, the expression becomes:

Simplify and evaluate the limit:

becomes:

Hence, the correct answer is Option 3.

Calculation:

⇒ f(π/2) = µ 

For continuous function ⇒ e^2/3 = e^λ = µ 

Now, 6λ + 6log_eµ + µ⁶ – e^6λ = 10

Hence, the correct answer is Option 4. 

Answer (1)

Sol.

To get the finite value,

1 + a – b = 0 

⇒ 

Apply L Hospital 

To get the finite value, a = –1

Also from (1)

b = 0 

∴ a = b = -1

Calcu;ation: 

⇒ 

= 

Hence, the correct answer is Option 3.

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,