EVALUATING LIMITS BY RATIONALIZING
Evaluating limits can be done in the following ways.
- Applying the given limit directly. While doing so, sometimes there is a chance to get indeterminant forms.
Some of the indeterminant forms are :
0/0, ∞/∞, ∞^{∞,} 0∞
- After simplification, we can apply the limit.
- After performing rationalization.
Conjugate of √a+b is √a-b
Conjugate of √a+√b is √a-√b
Conjugate of b+√a is b - √a
To perform rationalization, we have to multiply numerator and denominator by the conjugate.
Evaluating the given limits :
Problem 1 :
Solution:
=y→05+y-5y=y→05+y-55+y+5y5+y+5=y→05+y-5y5+y+5=y→0yy5+y+5=y→015+y+5=15+5=125×55=510
Problem 2 :
x→0x+3-3x
Solution:
=x→0x+3-3x=x→0x+3-3x+3+3xx+3+3=x→0x+3-3xx+3+3=x→01x+3+3=13+3=123×33=36
Problem 3 :
x→02x+1-1x
Solution:
=x→02x+1-1x=x→02x+1-12x+1+1x2x+1+1=x→02x+1-1x2x+1+1=x→022x+1+1=22(0)+1+1=22=1
Problem 4 :
x→-3x+7-2x+3
Solution:
=x→-3x+7-2x+3=x→-3x+7-2x+7+2x+3x+7+2=x→-3x+7-4x+3x+7+2=x→-3x+3x+3x+7+2=x→-31x+7+2=1-3+7+2=14+2=12+2=14
Problem 5 :
x→45-x-12-x
Solution:
=x→45-x-12-x=x→45-x-12-x×2+x2+x=x→45-x-12+x(4-x)×5-x+15-x+1=x→4(5-x-1)2+x(4-x)5-x+1=x→4(4-x)2+x(4-x)5-x+1=x→42+x5-x+1=2+45-4+1=2+21+1=42=2
Problem 6 :
x→41x-2-4x-4
Solution:
lim x→41x-2-4x-4=lim x→41x-2×x+2x+2-4x-4=lim x→4x+2x-4-4x-4=lim x→4x+2-4x-4=lim x→4x-2x-4=lim x→4x-2x-4×x+2x+2=lim x→4x-4(x-4)x+2=lim x→41x+2Applying the limit, we get=14+2=14
Problem 7 :
x→02x3-x+9
Solution:
=x→02x3-x+9=x→02x3+x+93-x+93+x+9=x→02x3+x+99-x-9=x→02x3+x+9-x=x→0-23+x+9=-23+0+9=-2(3+3)=-12
Problem 8 :
x→0x2+9-2x2+93x2+4-2x2+4
Solution:
=x→0x2+9-2x2+93x2+4-2x2+4=x→0x2+9-2x2+93x2+4-2x2+4×x2+9+2x2+93x2+4+2x2+4×3x2+4+2x2+4x2+9+2x2+9=x→0x2+9-2x2-93x2+4-2x2-4×3x2+4x2+9×2x2+42x2+9=x→0-x23x2+4+2x2+4x2x2+9+2x2+9=-0+4+0+40+9+0+9=-2+23+3=-46=-23
Problem 9 :
x→0+1x-1x2+x
Solution:
lim x→0+1x lim x→0+1x lim x→0+1x lim x→0+1x lim x→0+1x lim x→0+1x lim x→0+1x lim x→0+1x lim x→0+
Problem 10 :
x→016+x-4x
Solution:
=x→016+x-4x=x→016+x-416+x+4x16+x+4=x→016+x-16x16+x+4=x→0xx16+x+4=x→0116+x+4=116+4=14+4=18