Use taylor series to evaluate the limit select all...

Use Taylor series to evaluate the limit. SELECT ALL APPLICABLE CHOICES lim_(x->0)(tan^(-1)(6x)-tan(6x))/(x^(3)) A) -72 B) -12 C) -144 D) -54 Use Taylor series to evaluate the limit SELECT ALL APPLICABLE CHOICES tan-1(6x)-tan(6x) lim 0 x3 A) 72 B) 12 C) 144 D) 54

Transcript

00:01 In this exercise we are going to compute limit as x goes to 0 of f of x over g of x, where f of x is 1 plus ln of 1 plus 6x squared minus cosine of 6x, and g of x is x squared.

00:23 Before we go on, let's compute some derivatives of f of x.

00:27 Well, f prime of x is just 12x over 1 plus 6x squared, the derivative of this guy, plus sine of 6x multiplied by 6, so just 6 sine of 6x, the derivative of this guy.

00:51 Now let's observe that the evaluation of this guy at 0, well, is just 0.

00:59 Perfect.

00:59 The evaluation of f at 0, well, the evaluation of f at 0 is 1 plus ln of 1, which is 0, minus 1, which is cosine of 0.

01:14 So f of 0 is 0.

01:16 Perfect.

01:17 These two equations here show that the taylor series of f centered at x equal to 0 is of this form.

01:30 We are going to have f double prime of 0 over 2 multiplied by x squared plus the third derivative of f at 0 over 3 factorial multiplied by x cubed and so on.

01:45 So basically, the taylor series of f is going to start with the second degree term.

01:53 Perfect.

01:54 And, well, at this point, what do we know? we know that limit as x goes to 0 of f of x over d of x, by using the taylor series of f and the taylor series of g, which is just g, well, this limit here is just going to be the coefficient of x squared here.

02:20 So f double prime of 0 over 2.

02:24 Because when we divide this taylor series by x squared, we are going to get f double prime of 0 over 2...

Question

Use Taylor series to evaluate the limit. SELECT ALL APPLICABLE CHOICES lim_(x->0)(tan^(-1)(6x)-tan(6x))/(x^(3)) A) -72 B) -12 C) -144 D) -54 Use Taylor series to evaluate the limit SELECT ALL APPLICABLE CHOICES tan-1(6x)-tan(6x) lim 0 x3 A) 72 B) 12 C) 144 D) 54

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