Evaluate the limit, if it exists. (If an answer does not exist, enter DNE.)
lim t^2-64/2t^2+17t+8
t→−8
Please show steps!
the expression is (t-8)(t+8) / (2t+1)(t+8)
That is, for all t ≠ -8, it is just (t-8)/(2t+1)
I think you can handle that limit, eh?
wolframalpha is your friend ...
www.wolframalpha.com/input/?i=lim(t+-%3E+-8)+(+t%5E2-64)%2F(2t%5E2%2B17t%2B8)
To evaluate the limit as t approaches -8 for the given expression: lim(t→−8) (t^2 - 64) / (2t^2 + 17t + 8), we can proceed as follows:
Step 1: Substitute -8 into the expression:
(t^2 - 64) / (2t^2 + 17t + 8) = ((-8)^2 - 64) / (2(-8)^2 + 17(-8) + 8)
= (64 - 64) / (2(64) - 136 + 8)
= 0 / (128 - 136 + 8)
= 0 / 0
Step 2: Simplify the expression.
A limit exists if we can simplify the expression to a nonzero value, or determine that it approaches positive or negative infinity.
However, in this case, we have an indeterminate form (0/0) which means that further simplification is needed.
Step 3: Factorize the expression if possible.
We can factorize the numerator and denominator to see if any terms cancel out.
Numerator:
t^2 - 64 = (t - 8)(t + 8)
Denominator:
2t^2 + 17t + 8 = (2t + 1)(t + 8)
Step 4: Simplify the expression using factored form.
(t^2 - 64) / (2t^2 + 17t + 8) = [(t - 8)(t + 8)] / [(2t + 1)(t + 8)]
Step 5: Cancel out common terms.
Since there is a common factor of (t + 8) in both the numerator and denominator, we can cancel it out.
[(t - 8)(t + 8)] / [(2t + 1)(t + 8)] = (t - 8) / (2t + 1)
Step 6: Substitute the value of t.
Substituting t = -8 into the simplified expression, we get:
(t - 8) / (2t + 1) = (-8 - 8) / (2(-8) + 1)
= (-16) / (-16 + 1)
= (-16) / (-15)
= 16/15
Therefore, the limit as t approaches -8 for the given expression is 16/15.
To evaluate the limit, we substitute the given value (-8) into the function and simplify.
Let's start by plugging in -8 into the function:
lim(t→-8) (t^2 - 64) / (2t^2 + 17t + 8)
So we have:
((-8)^2 - 64) / (2(-8)^2 + 17(-8) + 8)
Simplifying the numerator:
(64 - 64) / (2(-8)^2 + 17(-8) + 8)
0 / (2(64) - 136 + 8)
0 / (128 - 136 + 8)
0 / 0
Division by zero is undefined, so the limit does not exist in this case.
Therefore, the answer is DNE (Does Not Exist).