Limit and Continuity MCQ Quiz - Objective Question with Answer for Limit and Continuity - Download Free PDF

Concept:

• Limit: exists if left-hand and right-hand limits coincide.
• Continuity: is continuous at when .
• Differentiability: is differentiable at if exists.
• Derivative limit: Even if exists, may fail to exist due to oscillation.

Calculation:

Given,

⇒

⇒

⇒ limit exists

⇒

⇒ limit = value

⇒ continuous

⇒

⇒

⇒ differentiable at 0

⇒ For ,

⇒ as

⇒ oscillates

⇒ overall limit of as does not exist

∴ statements 1, 2, 3 and 4 are all true.

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:

Given,

The function is defined as:

We are tasked with finding:

Check the left-hand limit for continuity at x = -1 :

Check the right-hand limit for continuity at x = -1 :

Since the left-hand limit (L.H.S) and right-hand limit (R.H.S) are not equal, the function is discontinuous at x = -1 .

Check the differentiability at x = 1 :

The left-hand derivative at x = 1  is and the right-hand derivative at  x = 1  is which means the function is not differentiable at x = 1 

∴ The function is neither continuous at x = -1  nor differentiable at x = 1 . 

Hence, the correct answer is Option 4.

Calculation:

Given,

The function is defined as:

We are tasked with finding:

For  |x| 3, so the derivative is:

Now, compute the limit of the derivative as x to 0:

∴ The value of  is 0.

The correct answer is Option (c)

Concept:

Calculation:

= 

= 

Let 

If x → ∞ then t → 0

= 

= 8 × 1 

= 8 

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:

For a limit to exist, Left-hand limit and right-hand limit must be equal.

Calculations:

For a limit to exist Left-hand limit and right-hand limit must be equal.

|x| can have two values 

|x | = - x when x is negative 

|x| = x when x is positive.

 = 

​​ = 

Here, 

Hence, does not exist. 

Concept:

Definition:

• A function f(x) is said to be continuous at a point x = a in its domain, if  exists or or if its graph is a single unbroken curve at that point.
• f(x) is continuous at x = a ⇔ .

Formulae:

Calculation: 

Since f(x) is given to be continuous at x = 0, .

Also,  because f(x) is same for x > 0 and x

.

Concept:

• We say f(x) is continuous at x = c if

LHL = RHL = value of f(c)

i.e., 

Calculation:

            (a ϵ Real numbers)

∴ f(x) = f(a), So continuous at everywhere

Important tip:

Quadratic and polynomial functions are continuous at each point in their domain

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:

Definition:

• A function f(x) is said to be continuous at a point x = a in its domain, if  exists or or if its graph is a single unbroken curve at that point.
• f(x) is continuous at x = a ⇔ .

Calculation:

For x ≠ 0, the given function can be re-written as:

Since the equation of the function is same for x 0, we have:

= 

For the function to be continuous at x = 0, we must have:

⇒ K = .