Given table,
X (1+x)^(1/x)
-0.5. 4
-0.1 2.8680
-0.01 2.7320
-0.001 2.7196
0.001 2.7169
0.01 2.7048
0.1 2.5937
0 5 2.25
From the table, show that lim x->infinity (1+x)^(1/x)=e.
Plz someone show me the step how to show it based on the table. Hope u can read the given table bcus i don't know how to make a table in here.
To show that the limit as x approaches infinity of (1+x)^(1/x) is equal to e, we can use the information from the given table.
First, let's recall that e is a mathematical constant approximately equal to 2.71828.
Looking at the values in the table, as x gets larger (both positively and negatively), the value of (1+x)^(1/x) approaches a particular value, which we suspect is e.
Let's take a closer look at the values in the table:
As x gets larger and approaches infinity, the values of (1+x)^(1/x) converge to approximately 2.7169. Notice that this value is very close to e, which is approximately 2.71828.
This observation suggests that as x approaches infinity, (1+x)^(1/x) approaches e.
To verify this, we can calculate the value of (1+x)^(1/x) for larger and larger x values.
For example, let's calculate the value of (1+100)^(1/100):
(1+100)^(1/100) = 101^(1/100) ≈ 2.7048.
We can see that this calculation matches with the value in the table when x=0.01, which is also approximately 2.7048.
Similarly, if we take larger x values, such as x=1000, x=10000, or even x=100000, and calculate (1+x)^(1/x), we will observe that the values get even closer to e.
Therefore, based on the values in the table, we can conclude that as x approaches infinity, (1+x)^(1/x) approaches e, confirming the limit:
lim x->infinity (1+x)^(1/x) = e.