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I am a novice in calculus and have been learning differentiation lately. I read many books and saw you tube videos. There I noticed that to find the slope of a point in a curve,people take a point $P=(x+h,f(x+h))$ and determine the slope $\frac{f(x+h)-f(x)}{h}$. Now to find the slope more accurately, they take the point $P$ close to $(x,h)$ and say that it boils down to a tangent line. But the tangent by definition will intersect the curve at only one point,in that case shouldn't $h=0$? I know that gives indeterminate form and the people use limits by taking $h$ close to $0$. My question is how is that line a tangent? Even if it is close to $0$,it is still not a $0$,right? So that line still intersects. In that case isn't the derivative an approximation instead of an exact answer. I may be wrong,so please forgive me if I made any mistake.

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  • $\begingroup$ You are right when you say that the true tangent would correspond to $h=0$. All values of $h$ which are close but not equal to zero, on the other hand, yield a curve which is only close to the tangent in the vicinity of the point of differentiation. And it might intersect the curve in one or more points. Taking the limit $h\to0$ allows to compute the slope of the true tangent, while, as you say, computing the slope directly for $h=0$ would give a $0/0$ indeterminate form. $\endgroup$
    – popoolmica
    Nov 11, 2020 at 14:40
  • $\begingroup$ As @popoolmica indicated, you should understand limits in order to understand derivatives $\endgroup$ Nov 11, 2020 at 14:42
  • $\begingroup$ I mean I know that it means what f(x) approaches when x is close to that value,but as I mentioned that tangent is a not real tangent,is it right? $\endgroup$
    – a_i_r
    Nov 11, 2020 at 14:54
  • $\begingroup$ Maybe your confusion comes from the idea of an intersecting line that, so to say, "becomes" a tangent ( which is impossible, as you noted). The tangent is not the final state of this line when h is equal to O, but the " limiting position" ( as is ordinarily said) of the intersecting line. This line never reaches this limiting position, but it actually tends to it. So this limiting position actually exists ( and is, as I said, the tangent to the curve). $\endgroup$ Nov 11, 2020 at 16:04

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Relevant answer I wrote on PSE

The simple answer is that you are making the 'h' exactly to zero, you are going to down a chain of decreasing the size of the quantity till you can approximate the slope of tangent line.

Here is a simple exercise to understand this:

Give me any number, I will give you a number less than it. For example, you give me $.1$ then I will say $.01$, if you give me $.001$ then I can give you $.00001$ .. so it an endless limiting process. And we can keep having this exchange till we reach the $h$ value where the rate of change of the function with the input is more or less the slope of tangent line.

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