Evaluate the limit Lim (1 + (2/x))²x→∞
To evaluate the limit, we need to apply some algebraic manipulation and use the properties of limits
To evaluate the limit, we need to apply some algebraic manipulation and use the properties of limits.
First, let’s simplify the expression within the limit:
(1 + (2/x))^2
= (1 + (2/x)) * (1 + (2/x))
= 1 + 2/x + 2/x + 4/x^2
= 1 + 4/x + 4/x + 4/x^2
Now, let’s examine the individual terms:
1 approaches 1 as x approaches infinity.
The term 4/x becomes smaller as x approaches infinity since the denominator becomes larger. Hence, 4/x approaches 0.
Similarly, the term 4/x^2 becomes even smaller as x approaches infinity since the denominator becomes even larger. Hence, 4/x^2 approaches 0 as well.
Therefore, the expression simplifies to:
1 + 0 + 0 = 1
Hence, the limit as x approaches infinity of (1 + (2/x))^2 is equal to 1.
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