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.
actually is lim x->0 (1+x)^(1/x)=e. not lim x->infinity (1+x)^(1/x)=e.
To show that the limit as x approaches infinity of (1+x)^(1/x) is equal to e, we can observe the values in the given table.
As x approaches infinity, we can see that the value of (1+x)^(1/x) approaches 2.7183, which is approximately equal to the value of e.
Let's break down the steps to show this based on the table:
Step 1: Find the value of (1+x)^(1/x) as x approaches positive infinity.
From the table, as x becomes larger and positive (approaching infinity), we can see that the value of (1+x)^(1/x) becomes closer and closer to 2.7183. The values in the table such as 2.7169, 2.7048, and 2.5937 indicate this convergence towards 2.7183.
Step 2: Compare the value obtained in step 1 with the value of e.
The value of e is approximately 2.7183. We can observe that the values from the table are approaching this value as x increases infinitely. Thus, the limit as x approaches infinity of (1+x)^(1/x) is equal to e.
In summary, by examining the values in the given table, we can see that as x approaches positive infinity, the expression (1+x)^(1/x) converges towards the value of e, which is approximately equal to 2.7183.