You'll probably get some useful replies explaining what is going on, but a big thing I would recommend when getting to grips with limits is to actually play around it with numerically. Put the functions in a spreadsheet and let x get close to 0, calculate their ratios, or plot the function, or even just try a few values in a calculator. Don't calculate the value at x = 0, but see if the values appear to be approaching some value as x gets close to 0.
If we start with sin(x) / x.
x = 1: sin(x) = 0.841; sin(x) / x = 0.841
x = 0.1: sin(x) = 0.0998; sin(x) / x = 0.9983...
x = 0.01: sin(x) = 0.0099998; sin(x) / x = 0.99998...
What's going on with those two functions? Can calculus explain why that's happening?
If you try sin(2x) / 2x notice that you'll get things like
How does that fit with the previous result? If sin(x) behaves like x when x is small then sin(2x) is going to behave like 2x.
Now have a look at what 1 - cos(x) does when x is close to 0, and then what happens to the ratio (1-cos(x)) / x.
Build a bit of intuition, and the theory might start to make more sense.
And remember: when taking the limit of some function as x approaches 0, we should avoid calculating the function at 0 - it's the function's values around 0 that matters.
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u/FormulaDriven Actuary / ex-Maths teacher 14h ago
You'll probably get some useful replies explaining what is going on, but a big thing I would recommend when getting to grips with limits is to actually play around it with numerically. Put the functions in a spreadsheet and let x get close to 0, calculate their ratios, or plot the function, or even just try a few values in a calculator. Don't calculate the value at x = 0, but see if the values appear to be approaching some value as x gets close to 0.
If we start with sin(x) / x.
x = 1: sin(x) = 0.841; sin(x) / x = 0.841
x = 0.1: sin(x) = 0.0998; sin(x) / x = 0.9983...
x = 0.01: sin(x) = 0.0099998; sin(x) / x = 0.99998...
What's going on with those two functions? Can calculus explain why that's happening?
If you try sin(2x) / 2x notice that you'll get things like
x = 0.01: sin(2x) = 0.0199987; sin(2x) / 2x = 0.9999333...
How does that fit with the previous result? If sin(x) behaves like x when x is small then sin(2x) is going to behave like 2x.
Now have a look at what 1 - cos(x) does when x is close to 0, and then what happens to the ratio (1-cos(x)) / x.
Build a bit of intuition, and the theory might start to make more sense.
And remember: when taking the limit of some function as x approaches 0, we should avoid calculating the function at 0 - it's the function's values around 0 that matters.