Suppose limx→c f(x) = 5 and lim x→c g(x) = −2. Find


b. limx→c 2f(x)g(x)

Verified step by step guidance

1. First, understand that the problem involves finding the limit of a product of functions as x approaches a certain value, c.
2. Recall the limit property for products: if lim(x→c) f(x) = L and lim(x→c) g(x) = M, then lim(x→c) [f(x) * g(x)] = L * M.
3. In this problem, you are given that lim(x→c) f(x) = 5 and lim(x→c) g(x) = -2.
4. Apply the limit property for products to find lim(x→c) [2 * f(x) * g(x)]. This can be rewritten as 2 * lim(x→c) [f(x) * g(x)].
5. Substitute the known limits into the expression: 2 * (5 * -2). Calculate this expression to find the limit.

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→c f(x) = 5 indicates that as x gets closer to c, the function f(x) approaches the value 5. 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 can be found by multiplying the individual limits. Specifically, if limx→c f(x) = L and limx→c g(x) = M, then limx→c (f(x)g(x)) = L * M. This property is essential for solving the given limit problem involving the functions f(x) and g(x).

Constant Multiplication in Limits

When calculating limits, multiplying a function by a constant does not affect the limit's existence. Specifically, if k is a constant and limx→c f(x) = L, then limx→c (k * f(x)) = k * L. This concept is crucial for evaluating the limit limx→c 2f(x)g(x), as the constant 2 can be factored out when calculating the limit.