Algebraic Method for Finding the Limit of (2^h - 1)/h as h Approaches 0

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In summary, the limit (2^h - 1)/h as h approaches 0 can be evaluated algebraically by using the derivative of ex and L'Hospital's rule, resulting in the answer ln(2).
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Miike012
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How do you differentiate lim (2^h - 1)/h ?
h --> 0
I would like to know how to do it algebraically instead of picking values of x that approach 0 and plugging it into 2^x.
 
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Miike012 said:
How do you differentiate lim (2^h - 1)/h ?
h --> 0
I would like to know how to do it algebraically instead of picking values of x that approach 0 and plugging it into 2^x.

I presume you mean how do you evaluate that limit. If you know the derivative of ex you can do it this way:

[tex]\lim_{h\rightarrow 0}\frac{2^h-1}{h}=
\lim_{h\rightarrow 0}\frac{e^{h\ln 2}-1}{h}=
(\ln 2)\ \lim_{h\rightarrow 0}\frac{e^{h\ln 2}-1}{h\ln 2}[/tex]

Not let t = h ln(2) giving

[tex](\ln 2)\ \lim_{t\rightarrow 0}\frac{e^{t}-1}{t}[/tex]

That gives ln(2) as the answer because the limit of the fraction is 1. You can see that either by recognizing that difference quotient as the derivative of ex at x = 0 or by applying L'Hospital's rule to it if you have had that.
 

FAQ: Algebraic Method for Finding the Limit of (2^h - 1)/h as h Approaches 0

What is the purpose of the algebraic method for finding limits?

The algebraic method for finding limits is used to determine the value that a function approaches as the input approaches a specific value, in this case, as h approaches 0.

How is the algebraic method for finding limits different from other methods?

The algebraic method for finding limits involves manipulating the algebraic expression of the function without actually evaluating it at the specific value. This allows us to determine the limit without having to know the exact value of the function at the specific value.

What is the formula used for finding the limit using the algebraic method?

The formula used for finding the limit using the algebraic method is: lim (f(x)) = f(a), where a is the specific value that the input is approaching.

What are the steps involved in using the algebraic method for finding the limit?

The steps involved in using the algebraic method for finding the limit are:
1. Rewrite the expression as a fraction.
2. Factor out any common factors.
3. Cancel out any common factors between the numerator and denominator.
4. Substitute the specific value for the input.
5. Simplify the expression to obtain the limit.

What are the advantages of using the algebraic method for finding limits?

One advantage of using the algebraic method for finding limits is that it does not require the use of a graph or a table. It also allows for a more precise determination of the limit, as it does not rely on estimation or visual interpretation. Additionally, the algebraic method can be used for any type of function, whereas other methods may only work for specific types of functions.

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