Question

Please include the full solution of how to get to the e^ln(1+1/x)^x form before the hospital rule.

Answer 1

To express (1+1/x)^x as e^ln(1+1/x)^x, take the natural log of (1+1/x)^x, multiply by x, and then raise e to that power, utilizing the inverse relationship between e^x and ln(x).

Step-by-step explanation:

To reach the expression e^ln(1+1/x)^x before applying L'Hôpital's rule, we must recognize the relationship between exponential functions and logarithms. The exponential function e^x and the natural logarithm ln(x) are inverse functions. This means that they cancel each other out, so e^(ln(x)) = x and ln(e^x) = x.

Using these properties, we can manipulate the expression (1+1/x)^x by first taking the natural logarithm of it and then raising e to the power of that logarithm. Thus, we get:

• Step 1: Take the natural logarithm: ln((1+1/x)^x)
• Step 2: Use the properties of logarithms: x * ln(1+1/x)
• Step 3: Exponentiate with base e: e^(x * ln(1+1/x)), which simplifies to e^ln(1+1/x)^x

This form is particularly useful when applying limits or L'Hôpital's rule, as it simplifies the expression into a more manageable form for calculus operations.