evaluate the following limit...
((1-x)/((1/e)-(e^-x))) as x approaches one
To evaluate the given limit, we can use the concept of continuity.
Let's substitute the given value, x = 1, into the expression:
((1-x)/((1/e)-(e^-x)))
= ((1-1)/((1/e)-(e^-1)))
= (0/((1/e)-(1/e)))
Since the denominator is zero, we need to simplify the expression further.
First, we can simplify the denominator:
((1/e)-(1/e)) = 0
Thus, the expression becomes 0/0.
When we encounter an expression with a 0/0 form, it indicates an indeterminate form, and we can resolve it using algebraic manipulation or applying L'Hôpital's Rule.
To apply L'Hôpital's Rule, we differentiate the numerator and denominator with respect to x:
Differentiating the numerator:
d/dx (1-x) = -1
Differentiating the denominator:
d/dx((1/e)-(e^-x)) = 0 - (-e^-x)
= e^-x
Now, we have:
(-1)/e^-x
As x approaches 1, let's substitute this value back into the expression:
(-1)/e^-1
Now, we can simplify this further:
= -1/e
Therefore, the limit of ((1-x)/((1/e)-(e^-x))) as x approaches 1 is equal to -1/e.