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# Class 11 NCERT Solutions- Chapter 13 Limits And Derivatives – Exercise 13.1 | Set 2

• Last Updated : 07 Apr, 2021

### Question 17:

Solution:

In, as xâ‡¢0

As we know, cos 2Î¸ = 1-2sin2Î¸

Substituting the values, we get

=

Put x = 0, we get

As, this limit becomes undefined

Now, let’s multiply and divide the numerator by x2 and denominator byto make it equivalent to theorem.

Hence, we have

=

=

By using the theorem, we get

=

=

=

= 4

### Question 18:

Solution:

In, as xâ‡¢0

Put x = 0, we get

As, this limit becomes undefined

Now, let’s simplify the equation to make it equivalent to theorem.

Hence, we have

=

By using the theorem, we get

=

=

Putting x=0, we have

=

### Question 19:

Solution:

In, as xâ‡¢0

Put x = 0, we get

= 0 Ã—1

= 0

### Question 20:

Solution:

In, as xâ‡¢0

Put x = 0, we get

As, this limit becomes undefined

Now, let’s simplify the equation to make it equivalent to theorem.

Hence, we can write the equation as follows:

=

By using the theorem, we get

=

=

=

=

Putting x=0, we have

= 1

### Question 21:

Solution:

In, as xâ‡¢0

By simplification, we get

Put x = 0, we get

As, this limit becomes undefined

Now, let’s simplify the equation to make it equivalent to theorem:

By using the trigonometric identities,

cos 2Î¸ = 1-2sin2Î¸

sin 2Î¸ = 2 sinÎ¸ cosÎ¸

Hence, we can write the equation as follows:

=

=

Putting x=0, we have

= 0

### Question 22:

Solution:

In, as xâ‡¢

Put x =, we get

As, this limit becomes undefined

Now, let’s simplify the equation :

Let’s take

As, xâ‡¢â‡’ pâ‡¢0

Hence, we can write the equation as follows:

=

=(As tan (Ï€+Î¸) = tan Î¸)

=

=

Now, let’s multiply and divide the equation by 2 to make it equivalent to theorem

=

=

As pâ‡¢0, then 2pâ‡¢0

=

Using the theorem and putting p=0, we have

= 2Ã—1Ã—1

= 2

### Question 23: Findand, where

Solution:

Let’s calculate, the limits when xâ‡¢0

Here,

Left limit =

Right limit =

Limit value =

Hence,, then limit exists

Now, let’s calculate, the limits when xâ‡¢1

Here,

Left limit =

Right limit =

Limit value =

Hence,, then limit exists

### Question 24: Find, where

Solution:

Let’s calculate, the limits when xâ‡¢1

Here,

Left limit =

Right limit =

As,

Hence, limit does not exists when xâ‡¢1.

### Question 25: Evaluate, where

Solution:

Let’s calculate, the limits when xâ‡¢0

Here,

As, we know that mod function works differently.

In |x-0|, |x|=x when x>0 and |x|=-x when x<0

Left limit =

Right limit =

As,

Hence, limit does not exists when xâ‡¢0.

### Question 26: Find , where

Solution:

Let’s calculate, the limits when xâ‡¢0

Here,

As, we know that mod function works differently.

In |x-0|, |x|=x when x>0 and |x|=-x when x<0

Left limit =

Right limit =

As,

Hence, limit does not exists when xâ‡¢0.

### Question 27: Find, where f(x)=|x|-5.

Solution:

Let’s calculate, the limits when xâ‡¢5

Here,

As, we know that mod function works differently.

In |x-0|, |x|=x when x>0 and |x|=-x when x<0

Left limit =

Right limit =

Hence,, then limit exists

### Question 28: Supposeand ifwhat are possible values of a and b?

Solution:

As, it is given

Let’s calculate, the limits when xâ‡¢1

Here,

Left limit =

Right limit =

Limit value f(1) = 4

So, as limit exists then it should satisfy

Hence, a+b = 4 and b-a = 4

Solving these equation, we get

a = 0 and b = 4

### What is? For some a â‰  a1, a2, …, an, compute.

Solution:

Here, f(x) = (x-a1) (x-a2)………… (x-an).

Then,

=

= (a1-a1) (a1-a2)………… (a1-an)

= 0

Now, let’s calculate for

=

= (a-a1) (a-a2)………… (a-an)

= (a-a1) (a-a2)………… (a-an)

### For what value (s) of a doesexists?

Solution:

Here,

As, we know that mod function works differently.

In |x-0|, |x|=x when x>0 and |x|=-x when x<0

Let’s check for three cases of a:

• When a=0

Let’s calculate, the limits when xâ‡¢0

Left limit =

Right limit =

As,

Hence, limit does not exists when xâ‡¢0.

• When a>0

Let’s take a=2, for reference

Let’s calculate, the limits when xâ‡¢2

Left limit =

Right limit =

As,

Hence, limit exists when xâ‡¢2.

• When a<0

Let’s take a=-2, for reference

Let’s calculate, the limits when xâ‡¢ -2

Left limit =

Right limit =

As,

Hence, limit exists when xâ‡¢ -2.

### Question 31: If the function f(x) satisfies, evaluate

Solution:

Here, as it is given

Put x = 1 in RHS, we get

= 2

Hence proved!

### Question 32: If. For what integers m and n does bothandexists?

Solution:

Let’s calculate, the limits when xâ‡¢0

Here,

Left limit =

Right limit =

Hence,

, then limit exists

m = n

Now, let’s calculate, the limits when xâ‡¢1

Here,

Left limit =

Right limit =

Hence,, then limit exists.

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