Question

lim x->-∞ arctan( |9- 4x^2| / 3 - 4x^2 - 4x )

Answer 1

This is difficult to explain, so it is best to use an online service.

In google paste:

limit calculator emathhelp

When you see list of results click on:

Limit Calculator - eMathHelp

When page be open in rectangle

Enter a function

paste:

atan ( abs (9- 4x^2) /( 3 - 4x^2 - 4x) )

In rectangle Find the limit at

type

inf

then click option

CALCULATE

You will see solution step by step.

I'll give you an explanation of the last step in the calculation because it can be confusing the way it's written.

( 4 ( - 4 + 3 ( 0 ) )⁻¹ ) =

4 • ( - 4 + 3 • 0 )⁻¹ =

4 • ( - 4 + 0 )⁻¹ = 4 • ( - 4 )⁻¹

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Remark:
a⁻ⁿ = 1 / aⁿ

so

( - 4 )⁻¹ = 1 / - 4¹ = 1 / - 4 = - 1 / 4
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4 • ( - 4 )⁻¹ = 4 • ( - 1 ) / 4 = - 1

atan ( 4 ( - 4 + 3 ( 0 ) ) ⁻¹ ) =

atan ( - 1 ) = - π / 4

Answer 2

3 - 4x^2 - 4x = -(4x^2+4x-3) = -(2x-1)(2x+3)
so now we have
|(2x+3)(2x-3)|/(-(2x+3)(2x-1)) = -|2x-3|/(2x-1)
lim(x→-∞) = -1
and arctan(-1) = -π/4