SOLUTION: For each of the limits determine whether the limit exists as a number, as an infinite limit (-∞,∞) or does not exist. If the limit is a number, evaluate it.
(a) lim(
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Question 970712: For each of the limits determine whether the limit exists as a number, as an infinite limit (-∞,∞) or does not exist. If the limit is a number, evaluate it.
(a) lim(x→(1/2))
(b) lim(x→(0-))
I am brand new to Calculus and I have absolutely no idea how these would be solved. I don't know what a limit means. I know what sin is and sqrt but what the question is asking and the limit parts are throwing me off.
Please help!
Answer by Boreal(15235) (Show Source): You can put this solution on YOUR website!
If you look at the values of x at 1/2, not approaching it, the sin (2x-1)/8x-4 fraction becomes 0/0, or undefined.
I will try to give the answers and a brief explanation.
2 cos (2x-1)/8x at x=1/2 This is 2 cos 0/4 or 2/8 or 1/4 by L'Hopital's Rule.
The idea of the limit is that as one approaches the value, but doesn't reach it, what is happening to the function? At 0.60, the value is 0.199/8, or just about 0.25, at 0.55, the sin (.1)/4.4-4 =even closer 0.25
At 0.54 it is .even closer
etc.
In other words, the value exists and continues to exist, so long as the value does not equal 1/2. I don't show here, but the values approach 0.25.
With more work, one can show the same is happening from the other side, The graph keeps approaching the same limit from both sides, and it exists for every value until the limit itself is reached. It is worth doing this on a calculator to several decimal places. The limit exists, and I used L'Hopital's rule to show what it was, although calculations as one gets close to the limit show the same thing.
For the second one, the limit exists and is 0, since coming from the negative side, the 4th root of t does exist, and it continues to exist until 0 is reached.
You will find in calculus that we can take the tangent to a curve and show what the curve is doing at a specific point. We take smaller and smaller steps, and as the steps approach 0, the limit will be reached. There are a lot of good sites online that discuss limits, both practically and mathematically. For the first problem above, it is really good practice to take both values as you get to 0.51, 0.501, 0,5001, and so forth. You will see the value approach 0.25,even though at 0.5, it does not exist.
Hope that helps some.
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