Non-zero over zero limits
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We can then just look at the values of \(f(x)\): f(x) = sin(x)/xĪs the ratio has a limit of \(1\) we conclude that \(l(n)\) goes to \(2\pi\). Adding a single-purpose function, such as logspace, to our vocabulary, or having to wrestle with these order of operations issues and integer versus floating point issues, leave us suggesting the use of a comprehension for this task. The hs vector could be defined more succinctly through different ways: logspace(-1, -10, 10), or with (1/10).
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These are just powers of 1/10 approaching 0. Next, we define some values getting close to \(0\): hs = But does the function have a limit?įirst, we note that this function is an even function, so if you are being thorough and worrying about getting close from the left and the right, you need only check one side. Even though both \(\sin(x)\) and \(x\) are continuous functions everywhere, the function is not continuous at \(0\) due to division by \(0\). Let's look at a simple example, the limit of \(f(x) = \sin(x)/x\) as \(x\) approaches \(0\) (a function Euler investigated as early as 1735). Imagining how \(f(x)\) is getting close - if it is - to some \(L\) which may be unknown. It is this intuitive idea we can approach with the computer, though we shall see that on the computer using floating point representations there are fundamental limits to how close we can get to a value.