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wiz0r
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Homework Statement
lim x->0 ( (a^x - 1)/x )
Homework Equations
NA
The Attempt at a Solution
The professor told me that the answer to that limit is log(a), but why? I don't understand; can someone explain why?
gb7nash said:Do you know l'hopital's rule?
The limit of (a^x - 1)/x as x approaches 0 is equal to ln(a), where ln is the natural logarithm function.
Exploring this limit is important because it can help us understand the behavior of exponential functions and their rates of change at a specific point. It can also have applications in areas such as calculus, physics, and finance.
To find the limit, we can use the L'Hopital's rule or algebraic manipulation. L'Hopital's rule states that if we have a limit of the form f(x)/g(x) as x approaches 0, and both f(x) and g(x) approach 0 or infinity, then the limit is equal to the limit of the quotient of their derivatives. Algebraic manipulation involves factoring out an x from the numerator and denominator and then simplifying the expression.
The value of this limit tells us the slope of the tangent line to the exponential function at the point where x = 0. It also tells us the rate of change of the function at that specific point.
Yes, a must be a positive real number, as the natural logarithm function is only defined for positive numbers. Additionally, a cannot be equal to 1, as the limit would then be undefined.