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What is the limit of #(1+sec^3 x)/tan^2 x# as x approaches 180?

Answer 1

I assume that #x# is approaching #180^@# (not radians). I also assume that you want to find this limit without resorting to l'Hopital's rue.

#1=sec^3x# is a sum of two cubes:

#1=sec^3x = (1+secx)(1-secx+sec^2x)#

We also know that #tan^2x+1=sec^2x#, so #tan^2x = sec^2x-1#. And #sec^2x-1# is a difference of squares, so we can factor it too.

Rewrite #(1+sec^3x)/tan^2x = ((1+secx)(1-secx+sec^2x))/((secx+1)(secx-1)) =(1-secx+sec^2x)/(secx-1) #

Now evaluate the limit by substitution.

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Answer 2

Christopher Allers

The limit of (1+sec^3 x)/tan^2 x as x approaches 180 is 1.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.