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How do you simplify #(2/a)/(1/(a+6))#?

Answer 1

Avery Alexander

#(2/a)/(1/(a+6)) = 2+12/a#

with exclusion #a != -6#

Note that division is the same as multiplication by the reciprocal:

#(2/a)/(1/(a+6)) = (2/a)-:(1/(a+6))#

#color(white)((2/a)/(1/(a+6))) = (2/a)*((a+6)/1)#

#color(white)((2/a)/(1/(a+6))) = (2a+12)/a#

#color(white)((2/a)/(1/(a+6))) = 2+12/a#

Note that both the original expression and the simplified expression are undefined when #a=0#, since they both involve division by #0#.

By way of contrast, the original expression is undefined when #a=-6#, but the simplified expression is well defined.

So the original expression and simplified expression are not equivalent when #a=-6#. So we need to explicitly exclude the value #a=-6#.

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Answer 2

Eva Abramson

To simplify (2/a)/(1/(a+6)), you can multiply the numerator and denominator by the reciprocal of the denominator. This gives you (2/a) * ((a+6)/1). Simplifying further, you get (2(a+6))/a.

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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.