limits by rationalization
DESCRIPTION
We solve limits by rationalizing. This is the second technique you may learn after limits by factoring. We solve two examples step by step. Watch video: http://www.youtube.com/watch?v=8CtpuojMJzA More videos and lessons: http://www.intuitive-calculus.com/solving-limits.htmlTRANSCRIPT
![Page 1: Limits by Rationalization](https://reader033.vdocuments.mx/reader033/viewer/2022052316/5592882f1a28ab56468b4748/html5/thumbnails/1.jpg)
![Page 2: Limits by Rationalization](https://reader033.vdocuments.mx/reader033/viewer/2022052316/5592882f1a28ab56468b4748/html5/thumbnails/2.jpg)
![Page 3: Limits by Rationalization](https://reader033.vdocuments.mx/reader033/viewer/2022052316/5592882f1a28ab56468b4748/html5/thumbnails/3.jpg)
Example 1
![Page 4: Limits by Rationalization](https://reader033.vdocuments.mx/reader033/viewer/2022052316/5592882f1a28ab56468b4748/html5/thumbnails/4.jpg)
Example 1
Let’s try to find the limit:
![Page 5: Limits by Rationalization](https://reader033.vdocuments.mx/reader033/viewer/2022052316/5592882f1a28ab56468b4748/html5/thumbnails/5.jpg)
Example 1
Let’s try to find the limit:
limh→0
√a + h −
√a
h
![Page 6: Limits by Rationalization](https://reader033.vdocuments.mx/reader033/viewer/2022052316/5592882f1a28ab56468b4748/html5/thumbnails/6.jpg)
Example 1
Let’s try to find the limit:
limh→0
√a + h −
√a
h
We rationalize the expression.
![Page 7: Limits by Rationalization](https://reader033.vdocuments.mx/reader033/viewer/2022052316/5592882f1a28ab56468b4748/html5/thumbnails/7.jpg)
Example 1
Let’s try to find the limit:
limh→0
√a + h −
√a
h
We rationalize the expression.In this case by multiplying and dividing by the conjugate.
![Page 8: Limits by Rationalization](https://reader033.vdocuments.mx/reader033/viewer/2022052316/5592882f1a28ab56468b4748/html5/thumbnails/8.jpg)
Example 1
Let’s try to find the limit:
limh→0
√a + h −
√a
h
We rationalize the expression.In this case by multiplying and dividing by the conjugate.
limh→0
√a + h −
√a
h=
![Page 9: Limits by Rationalization](https://reader033.vdocuments.mx/reader033/viewer/2022052316/5592882f1a28ab56468b4748/html5/thumbnails/9.jpg)
Example 1
Let’s try to find the limit:
limh→0
√a + h −
√a
h
We rationalize the expression.In this case by multiplying and dividing by the conjugate.
limh→0
√a + h −
√a
h= lim
h→0
![Page 10: Limits by Rationalization](https://reader033.vdocuments.mx/reader033/viewer/2022052316/5592882f1a28ab56468b4748/html5/thumbnails/10.jpg)
Example 1
Let’s try to find the limit:
limh→0
√a + h −
√a
h
We rationalize the expression.In this case by multiplying and dividing by the conjugate.
limh→0
√a + h −
√a
h= lim
h→0
√a + h −
√a
h.
![Page 11: Limits by Rationalization](https://reader033.vdocuments.mx/reader033/viewer/2022052316/5592882f1a28ab56468b4748/html5/thumbnails/11.jpg)
Example 1
Let’s try to find the limit:
limh→0
√a + h −
√a
h
We rationalize the expression.In this case by multiplying and dividing by the conjugate.
limh→0
√a + h −
√a
h= lim
h→0
√a + h −
√a
h.
√a + h +
√a√
a + h +√a
![Page 12: Limits by Rationalization](https://reader033.vdocuments.mx/reader033/viewer/2022052316/5592882f1a28ab56468b4748/html5/thumbnails/12.jpg)
Example 1
Let’s try to find the limit:
limh→0
√a + h −
√a
h
We rationalize the expression.In this case by multiplying and dividing by the conjugate.
limh→0
√a + h −
√a
h= lim
h→0
√a + h −
√a
h.���
����*1√
a + h +√a√
a + h +√a
![Page 13: Limits by Rationalization](https://reader033.vdocuments.mx/reader033/viewer/2022052316/5592882f1a28ab56468b4748/html5/thumbnails/13.jpg)
Example 1
Let’s try to find the limit:
limh→0
√a + h −
√a
h
We rationalize the expression.In this case by multiplying and dividing by the conjugate.
limh→0
√a + h −
√a
h= lim
h→0
√a + h −
√a
h.
√a + h +
√a√
a + h +√a
![Page 14: Limits by Rationalization](https://reader033.vdocuments.mx/reader033/viewer/2022052316/5592882f1a28ab56468b4748/html5/thumbnails/14.jpg)
Example 1
Let’s try to find the limit:
limh→0
√a + h −
√a
h
We rationalize the expression.In this case by multiplying and dividing by the conjugate.
limh→0
√a + h −
√a
h= lim
h→0
√a + h −
√a
h.
√a + h +
√a√
a + h +√a
= limh→0
(√a + h
)2 − (√a)2
h(√
a + h +√a)
![Page 15: Limits by Rationalization](https://reader033.vdocuments.mx/reader033/viewer/2022052316/5592882f1a28ab56468b4748/html5/thumbnails/15.jpg)
Example 1
Let’s try to find the limit:
limh→0
√a + h −
√a
h
We rationalize the expression.In this case by multiplying and dividing by the conjugate.
limh→0
√a + h −
√a
h= lim
h→0
√a + h −
√a
h.
√a + h +
√a√
a + h +√a
= limh→0
(√a + h
)2 − (√a)2
h(√
a + h +√a) = lim
h→0
a + h − a
h(√
a + h +√a)
![Page 16: Limits by Rationalization](https://reader033.vdocuments.mx/reader033/viewer/2022052316/5592882f1a28ab56468b4748/html5/thumbnails/16.jpg)
Example 1
Let’s try to find the limit:
limh→0
√a + h −
√a
h
We rationalize the expression.In this case by multiplying and dividing by the conjugate.
limh→0
√a + h −
√a
h= lim
h→0
√a + h −
√a
h.
√a + h +
√a√
a + h +√a
= limh→0
(√a + h
)2 − (√a)2
h(√
a + h +√a) = lim
h→0
�a + h − �ah(√
a + h +√a)
![Page 17: Limits by Rationalization](https://reader033.vdocuments.mx/reader033/viewer/2022052316/5592882f1a28ab56468b4748/html5/thumbnails/17.jpg)
Example 1
Let’s try to find the limit:
limh→0
√a + h −
√a
h
We rationalize the expression.In this case by multiplying and dividing by the conjugate.
limh→0
√a + h −
√a
h= lim
h→0
√a + h −
√a
h.
√a + h +
√a√
a + h +√a
= limh→0
(√a + h
)2 − (√a)2
h(√
a + h +√a) = lim
h→0
�a + h − �ah(√
a + h +√a)
= limh→0
h
h(√
a + h +√a)
![Page 18: Limits by Rationalization](https://reader033.vdocuments.mx/reader033/viewer/2022052316/5592882f1a28ab56468b4748/html5/thumbnails/18.jpg)
Example 1
Let’s try to find the limit:
limh→0
√a + h −
√a
h
We rationalize the expression.In this case by multiplying and dividing by the conjugate.
limh→0
√a + h −
√a
h= lim
h→0
√a + h −
√a
h.
√a + h +
√a√
a + h +√a
= limh→0
(√a + h
)2 − (√a)2
h(√
a + h +√a) = lim
h→0
�a + h − �ah(√
a + h +√a)
= limh→0
�h
�h(√
a + h +√a)
![Page 19: Limits by Rationalization](https://reader033.vdocuments.mx/reader033/viewer/2022052316/5592882f1a28ab56468b4748/html5/thumbnails/19.jpg)
Example 1
Let’s try to find the limit:
limh→0
√a + h −
√a
h
We rationalize the expression.In this case by multiplying and dividing by the conjugate.
limh→0
√a + h −
√a
h= lim
h→0
√a + h −
√a
h.
√a + h +
√a√
a + h +√a
= limh→0
(√a + h
)2 − (√a)2
h(√
a + h +√a) = lim
h→0
�a + h − �ah(√
a + h +√a)
= limh→0
�h
�h(√
a + h +√a) = lim
h→0
1√a + h +
√a
![Page 20: Limits by Rationalization](https://reader033.vdocuments.mx/reader033/viewer/2022052316/5592882f1a28ab56468b4748/html5/thumbnails/20.jpg)
Example 1
Let’s try to find the limit:
limh→0
√a + h −
√a
h
We rationalize the expression.In this case by multiplying and dividing by the conjugate.
limh→0
√a + h −
√a
h= lim
h→0
√a + h −
√a
h.
√a + h +
√a√
a + h +√a
= limh→0
(√a + h
)2 − (√a)2
h(√
a + h +√a) = lim
h→0
�a + h − �ah(√
a + h +√a)
= limh→0
�h
�h(√
a + h +√a) = lim
h→0
1
����:
√a√
a + h +√a
![Page 21: Limits by Rationalization](https://reader033.vdocuments.mx/reader033/viewer/2022052316/5592882f1a28ab56468b4748/html5/thumbnails/21.jpg)
Example 1
Let’s try to find the limit:
limh→0
√a + h −
√a
h
We rationalize the expression.In this case by multiplying and dividing by the conjugate.
limh→0
√a + h −
√a
h= lim
h→0
√a + h −
√a
h.
√a + h +
√a√
a + h +√a
= limh→0
(√a + h
)2 − (√a)2
h(√
a + h +√a) = lim
h→0
�a + h − �ah(√
a + h +√a)
= limh→0
�h
�h(√
a + h +√a) = lim
h→0
1
����:
√a√
a + h +√a
=1
2√a
![Page 22: Limits by Rationalization](https://reader033.vdocuments.mx/reader033/viewer/2022052316/5592882f1a28ab56468b4748/html5/thumbnails/22.jpg)
Example 1
Let’s try to find the limit:
limh→0
√a + h −
√a
h
We rationalize the expression.In this case by multiplying and dividing by the conjugate.
limh→0
√a + h −
√a
h= lim
h→0
√a + h −
√a
h.
√a + h +
√a√
a + h +√a
= limh→0
(√a + h
)2 − (√a)2
h(√
a + h +√a) = lim
h→0
�a + h − �ah(√
a + h +√a)
= limh→0
�h
�h(√
a + h +√a) = lim
h→0
1
����:
√a√
a + h +√a
=1
2√a
![Page 23: Limits by Rationalization](https://reader033.vdocuments.mx/reader033/viewer/2022052316/5592882f1a28ab56468b4748/html5/thumbnails/23.jpg)
Example 2
![Page 24: Limits by Rationalization](https://reader033.vdocuments.mx/reader033/viewer/2022052316/5592882f1a28ab56468b4748/html5/thumbnails/24.jpg)
Example 2
limx→0
√1 + x − 1
x
![Page 25: Limits by Rationalization](https://reader033.vdocuments.mx/reader033/viewer/2022052316/5592882f1a28ab56468b4748/html5/thumbnails/25.jpg)
Example 2
limx→0
√1 + x − 1
x
We rationalize using the same method.
![Page 26: Limits by Rationalization](https://reader033.vdocuments.mx/reader033/viewer/2022052316/5592882f1a28ab56468b4748/html5/thumbnails/26.jpg)
Example 2
limx→0
√1 + x − 1
x
We rationalize using the same method.
limx→0
√1 + x − 1
x=
![Page 27: Limits by Rationalization](https://reader033.vdocuments.mx/reader033/viewer/2022052316/5592882f1a28ab56468b4748/html5/thumbnails/27.jpg)
Example 2
limx→0
√1 + x − 1
x
We rationalize using the same method.
limx→0
√1 + x − 1
x= lim
x→0
√1 + x − 1
x.
![Page 28: Limits by Rationalization](https://reader033.vdocuments.mx/reader033/viewer/2022052316/5592882f1a28ab56468b4748/html5/thumbnails/28.jpg)
Example 2
limx→0
√1 + x − 1
x
We rationalize using the same method.
limx→0
√1 + x − 1
x= lim
x→0
√1 + x − 1
x.
√1 + x + 1√1 + x + 1
![Page 29: Limits by Rationalization](https://reader033.vdocuments.mx/reader033/viewer/2022052316/5592882f1a28ab56468b4748/html5/thumbnails/29.jpg)
Example 2
limx→0
√1 + x − 1
x
We rationalize using the same method.
limx→0
√1 + x − 1
x= lim
x→0
√1 + x − 1
x.��
�����*1√
1 + x + 1√1 + x + 1
![Page 30: Limits by Rationalization](https://reader033.vdocuments.mx/reader033/viewer/2022052316/5592882f1a28ab56468b4748/html5/thumbnails/30.jpg)
Example 2
limx→0
√1 + x − 1
x
We rationalize using the same method.
limx→0
√1 + x − 1
x= lim
x→0
√1 + x − 1
x.
√1 + x + 1√1 + x + 1
![Page 31: Limits by Rationalization](https://reader033.vdocuments.mx/reader033/viewer/2022052316/5592882f1a28ab56468b4748/html5/thumbnails/31.jpg)
Example 2
limx→0
√1 + x − 1
x
We rationalize using the same method.
limx→0
√1 + x − 1
x= lim
x→0
√1 + x − 1
x.
√1 + x + 1√1 + x + 1
= limx→0
(√1 + x
)2 − 12
x(√
1 + x + 1)
![Page 32: Limits by Rationalization](https://reader033.vdocuments.mx/reader033/viewer/2022052316/5592882f1a28ab56468b4748/html5/thumbnails/32.jpg)
Example 2
limx→0
√1 + x − 1
x
We rationalize using the same method.
limx→0
√1 + x − 1
x= lim
x→0
√1 + x − 1
x.
√1 + x + 1√1 + x + 1
= limx→0
(√1 + x
)2 − 12
x(√
1 + x + 1) = lim
x→0
1 + x − 1
x(√
1 + x + 1)
![Page 33: Limits by Rationalization](https://reader033.vdocuments.mx/reader033/viewer/2022052316/5592882f1a28ab56468b4748/html5/thumbnails/33.jpg)
Example 2
limx→0
√1 + x − 1
x
We rationalize using the same method.
limx→0
√1 + x − 1
x= lim
x→0
√1 + x − 1
x.
√1 + x + 1√1 + x + 1
= limx→0
(√1 + x
)2 − 12
x(√
1 + x + 1) = lim
x→0
�1 + x − �1x(√
1 + x + 1)
![Page 34: Limits by Rationalization](https://reader033.vdocuments.mx/reader033/viewer/2022052316/5592882f1a28ab56468b4748/html5/thumbnails/34.jpg)
Example 2
limx→0
√1 + x − 1
x
We rationalize using the same method.
limx→0
√1 + x − 1
x= lim
x→0
√1 + x − 1
x.
√1 + x + 1√1 + x + 1
= limx→0
(√1 + x
)2 − 12
x(√
1 + x + 1) = lim
x→0
�1 + x − �1x(√
1 + x + 1)
= limx→0
x
x(√
1 + x + 1)
![Page 35: Limits by Rationalization](https://reader033.vdocuments.mx/reader033/viewer/2022052316/5592882f1a28ab56468b4748/html5/thumbnails/35.jpg)
Example 2
limx→0
√1 + x − 1
x
We rationalize using the same method.
limx→0
√1 + x − 1
x= lim
x→0
√1 + x − 1
x.
√1 + x + 1√1 + x + 1
= limx→0
(√1 + x
)2 − 12
x(√
1 + x + 1) = lim
x→0
�1 + x − �1x(√
1 + x + 1)
= limx→0
�x
�x(√
1 + x + 1)
![Page 36: Limits by Rationalization](https://reader033.vdocuments.mx/reader033/viewer/2022052316/5592882f1a28ab56468b4748/html5/thumbnails/36.jpg)
Example 2
limx→0
√1 + x − 1
x
We rationalize using the same method.
limx→0
√1 + x − 1
x= lim
x→0
√1 + x − 1
x.
√1 + x + 1√1 + x + 1
= limx→0
(√1 + x
)2 − 12
x(√
1 + x + 1) = lim
x→0
�1 + x − �1x(√
1 + x + 1)
= limx→0
�x
�x(√
1 + x + 1) = lim
x→0
1√1 + x + 1
![Page 37: Limits by Rationalization](https://reader033.vdocuments.mx/reader033/viewer/2022052316/5592882f1a28ab56468b4748/html5/thumbnails/37.jpg)
Example 2
limx→0
√1 + x − 1
x
We rationalize using the same method.
limx→0
√1 + x − 1
x= lim
x→0
√1 + x − 1
x.
√1 + x + 1√1 + x + 1
= limx→0
(√1 + x
)2 − 12
x(√
1 + x + 1) = lim
x→0
�1 + x − �1x(√
1 + x + 1)
= limx→0
�x
�x(√
1 + x + 1) = lim
x→0
1
�����:
√1√
1 + x + 1
![Page 38: Limits by Rationalization](https://reader033.vdocuments.mx/reader033/viewer/2022052316/5592882f1a28ab56468b4748/html5/thumbnails/38.jpg)
Example 2
limx→0
√1 + x − 1
x
We rationalize using the same method.
limx→0
√1 + x − 1
x= lim
x→0
√1 + x − 1
x.
√1 + x + 1√1 + x + 1
= limx→0
(√1 + x
)2 − 12
x(√
1 + x + 1) = lim
x→0
�1 + x − �1x(√
1 + x + 1)
= limx→0
�x
�x(√
1 + x + 1) = lim
x→0
1
�����:
√1√
1 + x + 1
=1
2
![Page 39: Limits by Rationalization](https://reader033.vdocuments.mx/reader033/viewer/2022052316/5592882f1a28ab56468b4748/html5/thumbnails/39.jpg)
Example 2
limx→0
√1 + x − 1
x
We rationalize using the same method.
limx→0
√1 + x − 1
x= lim
x→0
√1 + x − 1
x.
√1 + x + 1√1 + x + 1
= limx→0
(√1 + x
)2 − 12
x(√
1 + x + 1) = lim
x→0
�1 + x − �1x(√
1 + x + 1)
= limx→0
�x
�x(√
1 + x + 1) = lim
x→0
1
�����:
√1√
1 + x + 1
=1
2
![Page 40: Limits by Rationalization](https://reader033.vdocuments.mx/reader033/viewer/2022052316/5592882f1a28ab56468b4748/html5/thumbnails/40.jpg)