I can’t figure them out. Pls help
In: Mathematics
Consider y=1/x. Simple math can tell us that no matter how large of a value we use for x, y will never equal zero.
But, we could say that, “as x approaches infinity, y approaches zero”. That’s pretty much what a limit statement sounds like out loud. We know it’ll never be zero, but that’s where that function is always headed, closer and closer to zero.
These are important because they let mathematicians work with infinitely small or infinitely large intervals and series. That leads to things like derivatives.
Imagine a curved line, and wanting to measure its “curvature” or its “rate of change”. You could hold a ruler across two points of that curve, and you might get a kind-of-sort-of idea of it. You might get a decent approximation of it, even. Now use two points that are closer to each other. Now use two points that are even closer. And closer. And closer.
See a pattern? If you “could” get infinitely close to a single point, and form a line that represents the precise “instantaneous” rate of change of the curve you’re measuring, you’d need a concept like a limit. If you were able to put a formula to calculate the rate of change of your original function for a given value, you’d have a derivative.
a limit is the point that y approaches when x approaches a specified value
or in more formal notation:
a limit is the point that f(x) approaches when x approaches a specified value
the derivative is itself a limit. you should look up the definition for it right now. to understand how the derivative works, look up secant lines then look up tangent lines. i promise you, this is the easiest way to understand derivatives (and the right way)
if you draw a secant line that crosses through two points on a curve, you have a rough estimate of the slope of the curve. since the function we are concerned with is a curve, the slope is different at each point on the curve and we can do better than a rough estimate. that’s where derivatives come in
imagine two points on a curve with one fixed and the other free to move. if you move the second point, you get a new secant line each time. if you move the point right on top of the fixed point, you get the line tangent to the curve at that point. the tangent line shares the same slope as the curve at that point
the process of moving the point is the same as saying the distance between the two points’ x coordinates (h) approaches zero. that’s what the lim h->0 part is saying. the rest of it is just the formula for the slope of the secant line. but the limit of this formula is the slope of the line tangent to curve at that value of x
once you get that understanding, you can then think of the derivative of a function as a new function all its own. if you take the derivative of f(x), you get a new function f'(x). this new function is capable of returning the slope of f(x) for any given value of x
you can use the limit formula to derive f'(x) for any given f(x), provided it exists. i suggest you give it a shot by picking a function (try f(x) = x^2 ) and plugging in the right information into the derivative formula. then solve the limit. you will end up with the derivative of f(x) aka f'(x). the derivative of x^2 is 2x
there are faster ways to find derivatives of functions but the definition is very important to learn and understand
as the other guy said, a limit is simply the value of the function as it approaches a value. but it’s also important to note that a function may approach a different value depending on which side you approach from, indicated by – or + in the limit. eg as x -> 2+ would be approaching 2 from the right.
a derivative is just the slope of the line. however, limits are important in this because you need to know that a slope actually exists. if the line approaching from the left of a point has a different slope than it does approaching from the right, the point where the two sides meet will not have a real slope.
you can prove it has a consistent slope/derivative at a point, that it’s ‘differentiable’, by checking that limit of the slope is the same from both sides. and you can prove the points meet up at a point, that it’s ‘continuous’, by checking that the limit of the function itself is the same from both sides.
limits are also pretty important because they can be used to represent final states, like what happens when x goes to infinity or negative infinity
overkill explanation, but if you’re starting to learn calculus this is the foundation level stuff that’s important to understanding what’s going on.
A limit is the value of the function as it approaches a given input. Simple example is limit as f(x) = 2x -> 2 would just be f(2) = 2(2) = 4.
A derivative is the slope of a function. Again taking f(x) = 2x which is just a straight line with a slope of 2 the derivative of it is just 2. For a more complex function like f(x)= x^2 the derivative is no longer just a single number because the slope of the this function changes so it derivative is 2x.
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