Final answer:
The question tackles evaluating two math limits involving indeterminate forms, where L'Hopital's rule might be applied. The first limit involves a difference of two functions as x approaches zero, and the second involves the behavior of an exponential function as c approaches zero from the positive side.
Step-by-step explanation:
The question involves evaluating two separate limits, both of which may involve indeterminate forms that require special techniques like L'Hopital's rule to solve. Let's break down each part:
Part a
Evaluating the limit lim x→0 (1/ln(x+1) -1/x ), we notice an indeterminate form of type ∞ - ∞ as x approaches 0. To resolve this, we can use L'Hopital's rule after finding a common denominator and differentiating the numerator and denominator. By doing so, the indeterminate form can be resolved, and we can find the limit.
Part b
For the limit lim c→0^+ mg/c (1-e^-ct/m), we also encounter an indeterminate form of type 0/0 when c approaches 0. In this case, we can apply L'Hopital's rule by taking the derivative of the numerator and denominator with respect to c. By evaluating these new expressions at c=0, we can find the limit.
L'Hopital's rule may be required in both cases, which is a technique used to evaluate the limits involving indeterminate forms. Additionally, understanding the behavior of functions near asymptotes is crucial when discussing limits and evaluating them at points where the function may not be defined.