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

Concept:

​A function f(x) is continuous at x = a, if  f(x) = f(x) = f(a).

Calculation:

Given: f(x) = \frac{\pi}{2}\end{array}\right.\)

f() = m ×  + 1

Left-hand limit = 

Applying the limits:

Left- hand limit = m ×  + 1

Right-hand limit = 

Applying the limits:

Right-hand limit = 1 + n

For the function to be continuous at x = ,

Left-hand limit = Right-hand limit = f(π/2)

⇒ m×  + 1 = 1 + n

⇒ n = 

The correct answer is n =  .

Concept:

If a function is continuous at a point a, then

sin(∞) = a, Where -1≤ a ≤ 1

Calculation:

Given:

f(0) = 1

f(x) = x sin (1/x)

Checking continuity at x = 0

L.H.L

= 

= 0 × sin(∞)

= 0 

R.H.L

= f(0) = 1

L.H.L ≠ R.H.L

Hence, function is discontinuous at x = 0.

Concept:

If a function is continuous at x = a, then L.H.L = R.H.L = f(a).

Left hand limit (L.H.L) of f(x) at x = a is  

Right hand limit (R.H.L) of f(x) at x = a is 

Calculation:

Concept:

The function f(x) is continuous at x = a if

 f(a-) = f(a) = f(a+)

Calculation:

Given, f(x) = |x| 

For x ≥ 0, f(x) = x

and for x

So function is continuous for x > 0 and x

At x = 0, 

f(0-) = f(0) = f(0+) = 0

⇒ f(x) is continuous at x = 0

∴ The correct answer is option (1).

Concept:

The function f(x) is continuous at x = a if

 f(a-) = f(a) = f(a+)

Calculation:

Given, f(x) = |x| 

For x ≥ 0, f(x) = x

and for x

So function is continuous for x > 0 and x

At x = 0, 

f(0-) = f(0) = f(0+) = 0

⇒ f(x) is continuous at x = 0

∴ The correct answer is option (1).

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:

• 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:

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:​

• .
• .
• .
• .

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 = .