For i) you just state the definition of coth(x).stuart4512 said:
A series expansion is a mathematical technique used to approximate a function by expressing it as an infinite sum of simpler functions. It is particularly useful for calculating values of functions that cannot be easily evaluated by other methods.
In calculus, series expansions are used to approximate the behavior of a function near a specific point. This allows us to find the derivatives and integrals of the function at that point, as well as other important properties such as convergence and divergence of the function.
L'Hopital's rule is a mathematical theorem that states that under certain conditions, the limit of a quotient of two functions is equal to the limit of the quotient of their derivatives. It is often used in conjunction with series expansion to evaluate indeterminate limits involving functions that cannot be easily evaluated directly.
The two conditions for using l'Hopital's rule are that the functions involved must be differentiable and that the limit of the quotient of their derivatives must exist. In addition, the limit of the original function must be of the form 0/0 or ∞/∞.
No, l'Hopital's rule can only be used for the indeterminate forms 0/0 and ∞/∞. It cannot be used for other indeterminate forms such as 1^∞ or ∞ - ∞, as these do not satisfy the conditions for the rule to be applied.
