To estimate: The value of using graph and then find the exact value of the same using L’Hospital’s rule.
The exact value of .
Use online graphing calculator and draw the graph of as shown below in Figure 1.
From Figure 1 it is identified that curve approaches approximately 7 as x approaches ∞.
Find the exact value with the help of L’Hospital’s rule as follows.
Let, . (1)
Take natural logarithm on both sides,
Therefore, . (2)
Obtain the value of the function as x approaches .
As x approaches , the numerator is,
As x approaches , the denominator is,
Thus, is in an indeterminate form.
Therefore, apply L’Hospital’s Rule and obtain the limit.
So, the equation (2) becomes,
Thus, the exact value of .
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