Evaluating the Limit: lim(x→2)((3^(x)+3^(3-x)-12)/(3^(3-x)-3^(x/2)))
In this article, we will evaluate the limit of a complex function as x approaches 2. The function is given by:
(3^(x)+3^(3-x)-12)/(3^(3-x)-3^(x/2))
To evaluate this limit, we will use various techniques and properties of limits.
Step 1: Simplify the Function
Before evaluating the limit, let's simplify the function by combining the terms with the same base (3):
(3^(x) + 3^(3-x) - 12) / (3^(3-x) - 3^(x/2))
= (3^(x) + 3^(3-x) - 3^2) / (3^(3-x) - 3^(x/2))
= (3^(x) + 3^(3-x) - 9) / (3^(3-x) - 3^(x/2))
Step 2: Evaluate the Limit
Now, let's evaluate the limit as x approaches 2:
lim(x→2) ((3^(x) + 3^(3-x) - 9) / (3^(3-x) - 3^(x/2)))
= ((3^2 + 3^(2-2) - 9) / (3^2 - 3^(2/2)))
= ((9 + 3^0 - 9) / (9 - 3^1))
= (3^0 / (9 - 3))
= 1 / 6
Therefore, the value of the limit is 1/6.
Conclusion
In this article, we evaluated the limit of a complex function as x approaches 2 using various techniques and properties of limits. The final answer is 1/6. This example demonstrates the importance of simplifying the function before evaluating the limit.
