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# Math Examples of Limits in calculus

• ## If f(x) = limx→2 (x2+2x+1)2?

Step1: First we write the given expression:  Lim x→2 (x2 +2x +1)2

Step 2:  Write the expression as given below :

Lim x→2 (x2 +2x +1)= [Lim x→2 (x2 +2x +1)] 2

Step 3: Now we individually solve this expression:

Lim x→2 (x2 +2x +1)= [Lim x→2 x2   + Lim x→2 2x + Lim x→2   1]2

Lim x→2 x2 = 4

Lim x→2 2x = 2 x 2 =4

Lim x→2   1= 1

Lim x→2 (x2 +2x +1)= (4+4+1)= 81

Step 4: Now we are plugging this expression together:

Lim x→2 (x2 +2x +1)= [Lim x→2 (x2 +2x +1)] 2

= 81

## Solve the given problem using power law of limit, f(x)=limx→2(3x)2 ?

According to power law,

If the limit limx→a ­f(x) exist, then

limx→a ­(f(x) )n=( limx→a ­f(x) )n

Now put the value of given function in the above formula of power Laws of Limit and firstly applying the limit and than using the power law solve the given function.

F(x) = limx→2(3x)2,

= (limx→2(3 x 2))2,

= (6)2=36.

Now, I am going to next example: the function is limx→1[x+2x]3, By using the power Law.

We have function limx→1[x+2x]3 by the law of power,

Putting the values in the above law of power and then applying the limit on the given function that is putting the value of x=1 in the given function, so, that the equation is,

= limx→1 [x+2x]3,

= limx→1[1+2]3,

=[3]3,

=27.

## Solving the problem with help of limit law of multiplication, Lim x→2 [(2x3) (3x) (x+1)]?

Step 1: Lim x→2 [(2x3) (3x) (x+1)],

Step 2:   Lim x→2 [(2x3) (3x) (x+1)] = [Lim x→2 2x3]  x  [Lim x→2 3x ]  x [  Lim x→2 (x+1)],

Step 3:  Lim x→2 2x3   = 16,

Lim x→2 3x= 3. Lim x→2 x    =   2 x 3 = 6,

Lim x→2 (x+1)=        Lim x→2 x    +  Lim x→2 1,

=     2 +1 =3,

Step 4:   Lim x→2 [(2x3) (3x) (x+1)] = [Lim x→2 2x3]  x  [Lim x→2 3x ]  x [  Lim x→2 (x+1)],

=16 x 6 x 3,

= 288

## Find the solution of the problem using multiplication law of limit, where expression is: Lim x→3 [(3x2 +1) (8x)]?

Step 1:   Lim x→3 [(3x2 +1) (8x)],

Step 2: Lim x→3 [(3x2 +1) (8x)] = Lim x→3   (3 x2 +1)    x Lim x→3  ( 8x),

Step 3: Lim x→3   ( 3 x +1)  =   3 Lim x→3    x2    +  Lim x→3   1,

=    3 x 9 +1 = 28,

Lim x→3   8x    = 8 Lim x→3   x   = 8 x 3 =24,

Step 4: Lim x→3 [(3x2 +1) (8x)] = Lim x→3    ( 3 x2 +1)    x  Lim x→3  ( 8x),

=   28 x 24,

= 672.

## Find the solution of the given problem using multiplication law of limit, where limit expression is: Lim x→3 [(x -5) (x +12)]?

Step 1: First we write the given expression:  Lim x→3 [(x -5) (x +12) ],

Step 2: Write the expression like:

Lim x→3 [(x -5) (x +12)] = Lim x→3 (x-5)    x   Lim x→3   (x+12),

Step 3: Now, individually solve this expression,

Lim x→3   (x-5)    = Lim x→3  x   -   Lim x→3    5,

=   3 - 5

=   - 2

Lim x→3   (x+12) = Lim x→3   x   +   Lim x→3   12

= 3 +12,

=   15,

Step 4: Now we plug-in this together expression,

Lim x→3 [(x -5) (x +12)] = Lim x→3 (x-5)    x   Lim x→3   (x+12),

=   -2 x 15,

= - 30.

## Solve the limit by division of law, where limit is, Lim x→3 (x3 + 23)/(x+2) (x2 -2x +4)?

Step 1: Write the given expression:

Lim x→3 (x+ 23)/(x+2) (x2 -2x +4),

Step 2 : Now we solve the numerator and denominator expression,

Lim x→3 (x+ 23)/(x+2) (x2 -2x +4)= lim x→3  [((x+2) (x2 -2x +4))/ ((x+2) (x2 -2x +4))],

= lim x→3  [(x+2)/(x+2)]

Step 3: Now we are simplify the getting expression:

lim x→3  [(x+2)/(x+2)]   =  lim x→3  1,

=   1,

Step 4: Then we plug-in the value in the given expression:

Lim x→3 (x+ 23)/(x-2) (x2 -2x +4) = 1.

## Solve the limit by division law, where limit is, Lim x→3 (x +3)/(x-1)?

Step 1: Write the given expression:

lim x→3  (x +3)/(x-1),

Step 2: Now we solve the numerator and denominator expression individually.

lim x→3 (x+3) =  6     ,  lim x→3  (x-1) = 2,

Step 3: Now we are simplify the getting expression:

lim x→3  (x +3)/(x-1) =   6/2 =3,

Step 4: Then we are plugging the value on the given expression:

lim x→3  (x +3)/(x-1) = 3.

## Solve the limit by division law, where limit is, Lim x→3 (x2 + 2x +1)/(x+1)?

Follow the below steps to solve the above mention problem:

Step 1: Write the given expression:

lim x→3  (x+ 2x +1)/(x+1),

Step 2: Now we solve the numerator and denominator expression

lim x→3  (x2   + 2x +1)/(x+1) = lim x→3  [((x+1) (x+1)) /(x+1)],

= lim x→3  (x+1),

Step 3: Now we are simplify the getting expression:

lim x→3  (x+1) =  lim x→3 x  + lim x→3 1,

=   3 +1 =4,

Step 4: Then we are plugging the value on the given expression:

Lim x→3 (x2   + 2x +1)/(x+1) = 4.

## Find the value of given function limx→0[1+4x] ,By using the subtraction Law?

The given function is:  limx→0[1+4x],

By the law of subtraction  limx→a[f(x)-g(x)]=( limx→af(x)- limx→a g(x)),

Putting the values in above equation and then applying the limit on the given function, here f(x) = limx→0(1) and g(x)= limx→0(4x) so, that the equation is,

= limx→0 (1)- limx→0(4x),

= limx→0(1-0),

=1.

## Find limx→1[2x+c], By using the subtraction Law?

We have limx→1[2x+c] by the law of subtraction,

limx→a[f(x)-g(x)]=( limx→af(x)- limx→a g(x)),

Plugging the values in above equation and then applying the limit on the given function,

= limx→1 (2x)- limx→1(c),

= limx→1(2-c),

=2-c.

## Find the value of given function limx→5[6x+x] ,By using the subtraction Law?

The given function is:  limx→5[6x+x],

By the law of subtraction  limx→a[f(x)-g(x)]=( limx→af(x)- limx→a g(x)),

Putting the values in above equation and then applying the limit on the given function, here f(x) = limx→5(6x) and g(x)= limx→5(x) so, that the equation is,

= limx→5 (6x)- limx→5(x),

= limx→5(30-5),

=25.

## Evaluate the limit limx→2[2x+7x] By using the subtraction Law?

We have limx→2[2x+7x] by the law of subtraction,

limx→a[f(x)-g(x)]=( limx→af(x)- limx→a g(x)),

Putting the values in above equation and then applying the limit on the given function, here f(x) = limx→2(2x) and g(x)= limx→2(7x) so,

= limx→2 (2x)- limx→2(7x),

= limx→2(4-14),

=-10.

## Evaluate the limit limx→3[x+5] By using the subtraction Law?

We have limx→3[x+5] by the law of subtraction,

limx→a[f(x)-g(x)]=( limx→af(x)- limx→a g(x)),

Putting the values in above equation and then applying the limit on the given function,

= limx→3 (x)- limx→3(5),

= limx→3(3-5),

=-2.

## Solve this question using addition law of limit, Lim x→4 (x + 2x +3)?

Lim x→4     (x  + 2x +3) = Lim x→4     (x ) + Lim x→4     ( 2x) + Lim x→4     (3),

Now here you can see we divided the limit into three  parts and we calculate both the parts our answer will be the same,

Now on solving on left hand side we will get:

Put x=4

4+2*4 +3

15,

Now  if talk about right hand side we will solve every limit separately then we will add,

Lim x→4 (x )=4,

Lim x→4 (2x)=2*4=8,

Third term has no x so its value remains as it was,

Now on combining we will get:

4+8+3=15,

Left hand side = right hand side,

## Solve this question using addition law of limit, Lim x→5 (x + 2)?

For solving this type of problem the basic need is to know what the addition law of limit is

Limit of any two Functions can be added if both the function having same limits.

Lim x→5 (x + 2) = Lim x→5     (x) + Lim x→5     ( 2),

Now here you can see we divided the limit into two parts and we calculate both the parts our answer will be the same

Firstly we will take the left hand side,

Lim x→5 (x  + 2),

On putting x=5 we will get the value of the function as 7,

Now we will take right hand side,

Lim x→5     (x ) + Lim x→5     ( 2),

If we solve first function we will get the value of the function as 5,

And when we solve the second function as it doesn’t have any x so the value of function remains as it is that is 2