Suppose limx→b f(x) = 7 and lim x→b g(x) = −3. Find


b. limx→b f(x)⋅g(x)

Verified step by step guidance

1. First, understand the concept of limits. A limit describes the value that a function approaches as the input approaches a certain point. In this problem, we are given the limits of two functions, f(x) and g(x), as x approaches b.
2. Next, recall the limit multiplication rule: if lim(x→b) f(x) = L and lim(x→b) g(x) = M, then lim(x→b) [f(x)⋅g(x)] = L⋅M. This rule allows us to multiply the limits of two functions directly.
3. Apply the limit multiplication rule to the given limits: lim(x→b) f(x) = 7 and lim(x→b) g(x) = -3. According to the rule, the limit of the product f(x)⋅g(x) as x approaches b is the product of the individual limits.
4. Calculate the product of the limits: multiply the limit of f(x), which is 7, by the limit of g(x), which is -3. This gives you the limit of the product f(x)⋅g(x) as x approaches b.
5. Finally, conclude that the limit of the product f(x)⋅g(x) as x approaches b is the result of the multiplication you performed in the previous step.

Key Concepts

Limit of a Function

The limit of a function describes the value that a function approaches as the input approaches a certain point. In this case, limx→b f(x) = 7 indicates that as x gets closer to b, the function f(x) approaches the value 7. Understanding limits is fundamental in calculus as it lays the groundwork for concepts like continuity and derivatives.

Product of Limits

The product of limits states that if the limits of two functions exist as x approaches a certain value, then the limit of their product is the product of their limits. Specifically, if limx→b f(x) = L and limx→b g(x) = M, then limx→b (f(x)⋅g(x)) = L⋅M. This property is essential for solving problems involving the multiplication of functions at a limit.

Evaluating Limits

Evaluating limits involves substituting values or applying limit laws to find the limit of a function as it approaches a specific point. In this scenario, to find limx→b f(x)⋅g(x), one would substitute the known limits of f(x) and g(x) into the product of limits formula, resulting in 7⋅(−3) = −21. This process is crucial for solving limit problems in calculus.