For ihe functions below that have removable discontinuity...

For the functions below that have removable discontinuity at x = a [if the function does not have a removable discontinuity, type "n"]; state the value of g(a) where g(x) agrees with f(x) for x ≠ a and is continuous everywhere below: (a) f(x) = (x^2 - 4) / (x + 2), a = -2 (b) f(x) = (x - 7) / (x^2 + 64), a = 7 (c) f(x) = (x^2 - 16) / (x + 4), a = -4 (d) f(x) = (x - 9) / (√x - 3), a = 9

Transcript

00:01 We are given with few functions and we need to find if the functions have the removable discontinuity at the given values of a.

00:11 And if s, we need to find the value of g of a where g of a agrees with f of x for x not equal to a.

00:22 In the first part, we are given with f of x is equal to x square minus 2x minus 8 divided by x plus.

00:31 2 and the value of a is equal to minus 2.

00:34 If we substitute x is equal to minus 2 in the denominator, the function will be undefined.

00:39 So let's compute the limit of f of x when x tends to minus 2.

00:46 It is equal to the limit when x tends to minus 2, the derivative of what is in the numerator and the derivative of what is in the denominator.

00:57 If we differentiate numerator, we will get 2x minus 2.

01:02 Differentiate the denominator, we will get 1.

01:05 So it will be equal to 2 times minus 2, which is equal to minus 6.

01:11 So what we can say is the value of g of a in this case is nothing but minus 6.

01:19 The function f of x has removable discontinuity at a is equal to minus 2.

01:25 Now let's move on to the part 2.

01:27 In part 2, we have f of x is equal to x minus 7 divided by modulus of x minus 7 we know that modulus of x minus 7 is not differentiable at x is equal to 7 why because the left -hand limit of f of x will be equal to minus 1 where the right -hand limit will be equal to 1 as both of them are not equal the function doesn't have removable discontinuity at x is equal to 7.

02:10 So this is the answer for the second part of the problem.

02:14 In the third part, we have f of x is equal to x cube plus 64 divided by x plus 4.

02:23 If we substitute x is equal to minus 4 in the denominator, it will be equal to 0...

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