Suppose f and g are continuous functions such that?

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Suppose f and g are continuous functions such that

g(4) = 5

and

lim

x → 4

[3f(x) + f(x)g(x)] = 32.

Find

f(4).

2 Answers

  • Since f and g are continuous, we have that

    lim f(x) = f(a)

    x → a

    lim g(x) = g(a)

    x → a

    for all real numbers a in R.

    Also, since f and g are continuous, so is 3f + fg so that

    lim [3f(x) + f(x)g(x)] = [3f(a) + f(a)g(a)]

    x → a

    for all real numbers a in R.

    In particular,

    lim [3f(x) + f(x)g(x)] = [3f(4) + f(4)g(4)]

    x → 4

    Since we are given that

    lim [3f(x) + f(x)g(x)] = 32

    x → 4

    and g(4) = 5, substituting these into the equation above, we have

    32 = [3f(4) + f(4)(5)]

    32 = 8f(4)

    4 = f(4).

    Thus f(4) = 4.

  • Thank you!!

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