Textbook Question
Finding Limits
In Exercises 9–24, find the limit or explain why it does not exist.
lim x→a (x² ― a²)/(x⁴ ― a⁴)
1. Limits and Continuity
Finding Limits Algebraically
1:40 minutesProblem 2.2.53
Textbook Question
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 guidance1
First, understand that the problem involves finding the limit of a product of functions as x approaches a certain value, c.
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.
In this problem, you are given that lim(x→c) f(x) = 5 and lim(x→c) g(x) = -2.
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)].
Substitute the known limits into the expression: 2 * (5 * -2). Calculate this expression to find the limit.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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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).
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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.
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