correct me if I'm wrong please.
1. What is the value of Lim 2x+3/4x-5 when x>00 (infinity?)
a.-3/5
b.0
c.1/2 <.. my answer
d.1
2. An object moves according to the law s = t^2 + 5t - 8. What is the average rate of change of s with respect to t in the interval t to (t + delta t )?
a.2t + 5
b.2t + 6
c.2t + delta t + 5 <... my answer
d.2t + (delta t )^2 +5 delta t
both correct
For the first question:
To find the value of the limit Lim (2x + 3)/(4x - 5) as x approaches infinity, we need to evaluate the expression when x gets very large.
First, divide both the numerator and denominator by x, as it is the highest power in the expression:
Lim [2x/x + 3/x]/[4x/x - 5/x]
This simplifies to:
Lim (2 + 3/x)/(4 - 5/x)
Now, as x approaches infinity, both 3/x and 5/x approach zero:
Lim (2 + 0)/(4 - 0)
This further simplifies to:
Lim 2/4
Therefore, the value of the limit is 1/2.
So, your answer c. 1/2 is correct.
For the second question:
To find the average rate of change of s with respect to t in the interval (t to t + delta t), you need to find the difference in s divided by the difference in t in that interval.
Let's calculate:
s1 = t^2 + 5t - 8 (when t = t)
s2 = (t + delta t)^2 + 5(t + delta t) - 8 (when t = t + delta t)
Now, subtract s1 from s2:
s2 - s1 = [(t + delta t)^2 + 5(t + delta t) − 8] - (t^2 + 5t - 8)
Expanding and simplifying:
s2 - s1 = t^2 + 2t(delta t) + (delta t)^2 + 5t + 5(delta t) - 8 - t^2 - 5t + 8
s2 - s1 = 2t(delta t) + (delta t)^2 + 5(delta t)
Divide this by the difference in t:
(s2 - s1)/(t + delta t - t) = [2t(delta t) + (delta t)^2 + 5(delta t)]/(delta t)
Cancel out the delta t:
Average rate of change = 2t + (delta t) + 5
We can see that the correct answer is b. 2t + 5.
Your answer, c. 2t + delta t + 5, is not correct because the delta t term should not be included in the expression for the average rate of change.