Find functions f and g such that lim x→1 f(x)=0 and lim x→1 (f(x)g(x))=5.

Key Concepts

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. In this case, we are interested in the limit of f(x) as x approaches 1, which is given to be 0. Understanding limits is crucial for analyzing the continuity and behavior of functions near specific points.

Product of Limits

The product of limits states that if the limits of two functions exist, the limit of their product can be found by multiplying the individual limits. However, if one of the limits is zero, as in lim x→1 f(x) = 0, we must carefully consider the behavior of the second function g(x) to achieve a non-zero limit for their product, specifically lim x→1 (f(x)g(x)) = 5.

Finding Functions

Finding functions that satisfy specific limit conditions often involves creative construction of functions. In this scenario, we need to identify functions f and g such that f approaches 0 while their product approaches 5. This may involve using functions that grow or decay in a controlled manner, such as f(x) = (x-1) and g(x) = 5/(x-1) to meet the limit requirements.