lim
x->0 (e^2x-1)/(e^x-1)
I know that the answer is 2, but how would I get 2?
e^2x = (e^x)^2
let z = e^x
then you have
(z^2 -1)/(z-1)
(z-1)(z+1) / (z-1)
so
z+1
e^x +1
e^0 + 1
1 + 1
about 2 all right
To find the limit of the function (e^2x-1)/(e^x-1) as x approaches 0, you can use algebraic manipulation and L'Hôpital's rule.
Step 1: Substitute the value of x into the expression:
(e^2(0)-1)/(e^0-1)
This simplifies to:
(1-1)/(1-1) = 0/0
Step 2: Apply L'Hôpital's rule. Differentiate the numerator and denominator separately with respect to x and find their limits as x approaches 0.
Differentiate the numerator:
d/dx (e^2x-1) = 2e^2x
Differentiate the denominator:
d/dx (e^x-1) = e^x
Take the limits of the differentiated numerator and denominator as x approaches 0:
lim x->0 2e^2x / e^x = 2
Therefore, the limit as x approaches 0 of (e^2x-1)/(e^x-1) is 2.
To find the limit of the function (e^(2x) - 1) / (e^x - 1) as x approaches 0, you can try direct substitution into the function, but it results in an indeterminate form of 0/0.
In such cases, it is helpful to simplify the expression by factoring out common factors or using algebraic manipulations. Here's how you can do it step by step:
1. Start with the given expression: (e^(2x) - 1) / (e^x - 1).
2. Notice that the numerator can be factored as a difference of squares: (e^x + 1)(e^x - 1).
3. Rewrite the expression: [(e^x + 1)(e^x - 1)] / (e^x - 1).
4. Cancel out the common factor of (e^x - 1) in the numerator and the denominator.
5. Simplify the expression further: e^x + 1.
Now, the expression becomes e^x + 1. To find the limit as x approaches 0, you can directly substitute the value of x into the expression:
lim(x->0) (e^x + 1) = e^0 + 1 = 1 + 1 = 2
Therefore, the limit of the function as x approaches 0 is indeed 2.