Product rule for limits
The product rule for limits is a theorem that allows us to find the limit of a product of two functions by taking the limit of each function separately
The product rule for limits is a theorem that allows us to find the limit of a product of two functions by taking the limit of each function separately.
Let’s say we have two functions, f(x) and g(x), and we want to find the limit of their product as x approaches a point, say c.
The product rule for limits states that if the limits of f(x) and g(x) exist as x approaches c, then the limit of the product of f(x) and g(x) also exists and is equal to the product of their limits. In other words:
lim┬(x→c)〖(f(x)g(x))=lim┬(x→c)〖f(x) 〗· lim┬(x→c)〖g(x)〗 〗
So, if lim┬(x→c)f(x)=L and lim┬(x→c)g(x)=M, then:
lim┬(x→c)〖(f(x)g(x))=L·M〗
It is important to note that the limits of the individual functions must exist for the product rule to apply. If either of the limits does not exist or is infinite, then the limit of the product may not exist or may be undefined.
The product rule for limits is a useful tool in calculus to simplify the process of finding limits of products of functions. It allows us to break down the limit of a product into simpler limits and evaluate them separately, making computations easier.
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