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Explanation:
To find the linear approximation, L(x), to the function f(x) = 1/x near x = a = 3, we can use the concept of linearization. The linear approximation can be represented as:
L(x) = f(a) + f'(a)(x - a)
First, let's find f(a) and f'(a).
f(a) = 1/a = 1/3
To find f'(a), we need to differentiate f(x) = 1/x. Using the power rule, we have:
f'(x) = -1/x^2
Substituting x = a = 3, we find:
f'(a) = -1/(3^2) = -1/9
Now, we can construct the linear approximation:
L(x) = f(a) + f'(a)(x - a)
= 1/3 - (1/9)(x - 3)
= 1/3 - (1/9)x + 1/3
Simplifying, we get:
L(x) = (2/3) - (1/9)x
Therefore, the linear approximation L(x) to f(x) = 1/x near x = 3 is given by L(x) = (2/3) - (1/9)x.