Limits properties

Some more examples of limits

Example # 1

Consider the problem of finding the limit for the following function when the value of x is greater than 1 :

The following table is drawn to illustrate the change in the value of the function when the value of n is 10. i.e., n > 0 and the value of x is greater than 1.

Value of x	Value of n	Value of xⁿ
100	10	100¹⁰
200	10	200¹⁰
1000	10	1000¹⁰

So, as x approaches

, xⁿ also approaches

. This can be simply expressed in the following form :

when n > 0 and x > 1.

Example # 2

Consider the problem of finding the limit for the following function when the value of x is greater than 1 :

The following table is drawn to illustrate the changes in the value of the functi on when the value of n is 2 . i.e., n > 0 and the value of x is greater than 0

Value of x	Value of n	Value of x^-n
10	2	0.01
100	2	0.001
1000	2	0.000001
10000	2	0.00000001

So, as x approaches

, x ^-n approaches 0. This can be simply expressed in the following form :

when n > 0 and x > 1.

Example # 3

Consider the problem of finding the limit for the following function when the value of x is greater than 1 :

The following table is drawn to illustrate the changes in the value of the function when the value of n is tending to

. i.e.,when n > 0 and n ->

and the value of x is greater than 0.

Value of x	Value of n	Value of xⁿ
100	2	100²
100	4	100⁴
100	10	100¹⁰
100	20	100²⁰

So, as n approaches

, x ⁿ approaches

. This can be simply expressed in the following form :

when n > 0 and x > 1.

Example # 4

Consider the problem of finding the limit for the following function when the value of x is greater than 1 :

The following table is drawn to illustrate the changes in the value of the function. when the value of n is tending to

. i.e.,when n > 0 and n ->

and the value of x is greater than 0.

Value of x	Value of n	Value of x^-n
100	2	0.0001
100	4	0.00000001
100	5	0.0000000001

So, as n approaches

, x ^-n approaches 0. This can be simply expressed in the following form :

when n > 0 and x > 1.

Example # 5

Consider the problem of finding the limit for the following function when the value of x is less than 1 and greater than 0.

The following table is drawn to illustrate the changes in the value of the function when the value of n approches

. i.e.,n > 0 and the value of x is less than 1 and greater than 0.

Value of x	Value of n	Value of xⁿ
0.01	2	0.0001
0.01	4	0.00000001
0.01	6	0.000000000001

So, as n approaches

, x^-n approaches 0. This can be simply expressed in the following form :

when n > 0 and 0 < x < 1.

Example # 6

Consider the problem of finding the limit for the following function when the value of x is less than 1 and greater than 0 :

The following table is drawn to illustrate the changes in the value of the function when the value of n is 2 . i.e., n > 0 and the value of x is less than 0.

Value of x	Value of n	Value of x^-n
0.01	2	10000
0.01	4	100000000
0.01	6	1000000000000

So, as n approaches

, x ^-n approaches

. This can be simply expressed in the following form

when n > 0 and 0 < x < 1.

Would you like to try these examples yourself?
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