lim t-->0 h(t), where h(t)= cos(t)-1/t^2.
for the table values t is +/- .002 , .0001 , .00005 , .00001
what do you guys get for the value of h(t) and limit??
when i plugged in .002 i got -.499999975
when i plugged in .0001 i got -.5
when i plugged in .00005 i got -.5
when i plugged in .00001 i got 0
so does the limit not exist since i got the value 0 for the last number??
Do you mean [cos(t)-1]/t^2 ?
or cos(t) - (1/t^2) ?
Is t in radians?
Only the first version has a finite limit.
If t is in radians, that limit is -1/2
You must have done the 0.00001 case incorrectly.
I assume the posted expression has missing parentheses, namely we're looking for:
lim t-->0 h(t),
where h(t)= ( cos(t)-1 )/t^2.
When dealing with limits, the calculator results are not always reliable, because there is only a limited number of digits in the working memory.
Your calculator displays to 9 significant digits, but incorrectly for the case of f(0.002), for which I get -0.499999833342
This means that for t=0.00001, it probably treats t² as zero, and the zero is a signal that something has gone wrong (like division by zero).
For t=0.00001, I get -0.50000004137019...
The limit is, in fact, -0.5. You can find it by using the d'Hôpital's rule:
Lim t→0 (cos(t)-1)/t²
=Lim (-sin(t))/(2t)
=Lim (-cos(t))/2
=-0.5
or by expanding cos(t) as a Taylor's series:
Lim t→0 (cos(t)-1)/t²
=Lim (1-t²/2+t4/4!-... -1)/t²
=Lim (-t²/2 + t4/24)/t²
=Lim (-(1/2) + t²/48 -...)
= -1/2
To find the limit of h(t) as t approaches 0, we can evaluate the function h(t) at the given table values and observe the behavior.
For t = +/- 0.002:
h(t) = cos(t) - 1/t^2
Plugging in t = 0.002, we get:
h(0.002) = cos(0.002) - 1/(0.002)^2 ≈ -0.499999975
Plugging in t = -0.002, we get:
h(-0.002) = cos(-0.002) - 1/(-0.002)^2 ≈ -0.499999975
For t = 0.0001:
h(0.0001) = cos(0.0001) - 1/(0.0001)^2 ≈ -0.5
For t = 0.00005:
h(0.00005) = cos(0.00005) - 1/(0.00005)^2 ≈ -0.5
For t = 0.00001:
h(0.00001) = cos(0.00001) - 1/(0.00001)^2 ≈ 0
Based on the values obtained, it seems that as t approaches 0, h(t) approaches different values from both sides. The limit of h(t) as t approaches 0 does not exist because the function does not approach a single value from both sides.
To find the limit of h(t) as t approaches 0, we need to evaluate h(t) as t gets closer and closer to 0. In this case, h(t) is given by h(t) = cos(t) - 1/t^2.
Let's evaluate h(t) for the given table values:
For t = +/- 0.002:
h(t) = cos(0.002) - 1/(0.002)^2
Using a calculator, we find that h(t) ≈ -0.49999875.
For t = 0.0001:
h(t) = cos(0.0001) - 1/(0.0001)^2
Using a calculator, we find that h(t) ≈ -0.5.
For t = 0.00005:
h(t) = cos(0.00005) - 1/(0.00005)^2
Using a calculator, we find that h(t) ≈ -0.5.
For t = 0.00001:
h(t) = cos(0.00001) - 1/(0.00001)^2
Using a calculator, we find that h(t) ≈ 0.
Based on the values obtained, we can see that as t approaches 0, the values of h(t) get closer and closer to -0.5. However, for t = 0.00001, we get h(t) = 0.
Therefore, we can conclude that the limit of h(t) as t approaches 0 exists and is equal to -0.5. The fact that h(t) takes the value 0 at a single point does not affect the existence of the limit.