b.Determine the lim x-->4 (x^2+x-20/8-2x)
What Im stuck on is this
f(4)= 4^2+4-20 / 8-2(4)
f(4)= 0/0
x^2+x-20/8-2x = (x-4)(x+5)/-2(x-4)
What do I do next I'm so confused ik i would eliminate the x-4 from numerator and denominator but what would i do with the -2 that belongs to (x-4) in denominator.
I feel lost??
Step1 for me in any limit question is actually sub in the approach value
If you get a real number, that's your answer,
all done!
so for x = 4, we get 0/0, ok, we got work to do
step: if you get 0/0 for simple rational expressions like the above, IT WILL FACTOR. Sometimes you have to do some fancy stuff first, but eventually it will factor.
For ours, it is simply
Step3: Factor, simplify and repeat Step1
lim x-->4 (x^2+x-20/8-2x) , x--->4
= lim (x+5)(x-4)/(2(4-x)) , notice x-4 and 4-x are opposites, so (x-4)/(4-x) = -1
= lim -1(x+5)/2 , x ---> 4
= -1(4+5)/2
= -4.5
Here is a cool trick for limits.
Pick a number very close to the approach value, e.g. x = 4.001
stick that in your calculator's memory
and evaluate the original expression
I got -4.5005
My answer is correct
I use this often before I start my algebra to predict what answer I should get.
Ok thanks that helped a lot!
so basically to get the negative 1 i just divide -4/4 correct?
careful:
what happened to your x's in
(x-4)/(4-x) ?
What really happened is this
(x-4)/(4-x)
= -1(4-x)/(4-x)
= -1[ (4-x)/(4-x) ]
= -1[ 1 ]
= -1
Any number divided by its opposite = -1
I can understand why you're feeling confused, but don't worry, I'm here to help you understand the steps!
So far, you have correctly factored the numerator of the expression, which is (x^2 + x - 20), as (x - 4)(x + 5). Now you need to simplify the denominator, which is (8 - 2x).
To simplify (8 - 2x), you can factor out a common factor of 2 from both terms. This gives you:
(8 - 2x) = 2(4 - x)
Now, you have the simplified expression:
(x - 4)(x + 5) / 2(4 - x)
At this point, you are correct that the numerator and denominator share the factor (x - 4). To simplify the expression further, you can cancel out this common factor, but you should be cautious with cancelling terms in an expression with a limit. Instead, you can rewrite the expression as:
[(x - 4)(x + 5)] / [2(4 - x)]
Now, you can see that the factor in the denominator also includes (4 - x), which is the same as -(x - 4) in the numerator. Therefore, you can rewrite the expression as:
-(x + 5) / 2
Now, you can take the limit of this simplified expression as x approaches 4:
lim(x->4) [-(x + 5) / 2]
Plugging in the value of x into the expression, you get:
-(4 + 5) / 2 = -9/2
So, the limit of the expression as x approaches 4 is -9/2.
I hope this explanation helps! Let me know if you have any further questions.