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Business Calculus

by Mallory
(Tallahassee, FL, USA)











































lim (12+4x)/((9/x^2)-1)
x-> -3

How do I solve this? I know I need to multiply by a common number, but I don't know what that common number is.

Comments for Business Calculus

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Aug 26, 2010
Business Calculus – Evaluate Limits
by: Staff

The question:


lim (12+4x)/((9/x^2)-1)
x-> -3

How do I solve this?
I know I need to multiply by a common number, but I don't know what that common number is.

The answer:

lim (12+4x)/((9/x^2)-1)
x-> -3


Remember the difference of squares formula from algebra
It is: a^2 – b^2 = (a + b)(a - b)

This is the same situation you have in the denominator of your limit problem
((9/x^2)-1) = (3/x + 1)(3/x – 1)

Now, lets solve the problem:

1. Factor the numerator: (12 + 4x) = 4(3 + x)
2. Factor the denominator: ((9/x^2)-1) = (3/x + 1)(3/x – 1)
3. this is what your problem looks like now:

4(3 + x) / (3/x + 1)(3/x – 1)

4. multiply your entire expression by the faction x/x : (note: x/x really equals the number 1, so you have not changed the value of your original expression when you multiply it by this fraction)


(4(3 + x) / (3/x + 1)(3/x – 1))*(x/x)


= 4(3 + x) * x/ (3/x + 1)(3/x – 1)*x

= 4x(3 + x) / (3/x + 1)(3/x – 1)*x

= 4x(3 + x) / (3x/x + x)(3/x – 1)

= 4x(3 + x) / (3 + x)(3/x – 1)


5. notice that the factor (3 + x) appears in both the numerator and denominator – these factors cancel

= 4x(3 + x) / (3 + x)(3/x – 1)

= 4x / (3/x – 1)

6. last step: substitute the limit value of -3 for the x, then evaluate the expression

= 4x / (3/x – 1)

= 4*(-3) / ((3/(-3)) – 1)

= -12 / (-1 – 1)

= -12 / (-2)

= 6 (a positive 6)


The final answer is:

lim (12+4x)/((9/x^2)-1) = 6
x-> -3


Thanks for writing.


Staff
www.solving-math-problems.com


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