If the function f(x) defined as f(x)={ 3 , x=0 (1+(a x+b x^3)/(x^2))^1 / x, x>0 .

step 1 In this problem, we are given that the function is continuous at x equals to 0, which means that

If the function f(x) defined as f(x)={ 3 , x=0 ...

If the function $f(x)$ defined as $f(x)=\left\{\begin{array}{c}3 \quad, x=0 \\ \left(1+\frac{a x+b x^{3}}{x^{2}}\right)^{1 / x}, & x>0\end{array}\right.$
is continuous at $x=0$, then
(A) $a=0$
(B) $b=e^{3}$
(C) $a=1$
(D) $b=\ln 3$

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Key Concepts

Continuity

Continuity requires that the value of a function at a point equals the limit of the function as the argument approaches that point. For a function to be continuous at a point, especially where the function is defined piecewise, the limit from one side must match the given function value at that point.

Exponential Limit (Definition of e)

The exponential limit, often expressed as lim??0 (1 + kx)^(1/x) = e?, is a fundamental concept in calculus that allows one to evaluate limits involving expressions raised to a power that tends to infinity. This concept is essential when rewriting indeterminate forms into a recognizable exponential form.

Handling Indeterminate Forms

When dealing with limits, expressions can sometimes take on indeterminate forms such as 1?. In these cases, algebraic manipulation and the application of known limit formulas help resolve the indeterminacy, ensuring a finite limit that can be matched to a desired function value for ensuring continuity.

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