Anonymous asked in Science & MathematicsMathematics · 1 month ago

Help with math pls?

The graph of the function f is shown in the xy-plane. 

The graph of f has a vertical asymptote at x=-2. The function g is continuous and increasing for all x. Values of g(x) at selected values of x are shown in the table above.

[Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number.](a) Using the graph of f and the table for g estimate limx→1(2f(x)+3g(x))(b) For each of the values a=-2, a=2, and a=3, determine whether or not f is continuous at x=a. In each case, use the definition of continuity to justify your answer.(c) Fine the value of limx→0(f(x)) or explain why the limit does not exist.(d) Write a difference quotient that best approximates the instantaneous rate of change of g at x=1.

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1 Answer

  • 1 month ago
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    It appears, that as x approaches 1, the limit of f(x) is 4 and the limit of g(x) is 5.

    lim(2f(x)+3g(x)) = 2limf(x) + 3limg(x) = 2*4 + 3*5

    When the limit from the left is not equal to the limit from the right, then the overall limit does not exist.

    A function is discontinuous at a point if the limit at that point does not exist or if that point is a hole.

    With g we don't know if it's continuous at x=1 so estimating the instantaneous rate of change does not make sense.

    However we could guess (5.004 - 4.996) / (1.001 - 0.999)

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