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I am a bit confused by the above statement from a textbook I am using. A tangent line is a line that touches a curve at a certain point, looking at $|x|$, it is not differentiable at 0 but isn't the x-axis tangent to this curve at 0? please help me understand.

mamotebang
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    Please add details of the textbook you are referring to: author, title, publisher and date. In isolation I'd have to guess about how a tangent is defined there. – hardmath Nov 04 '19 at 12:44
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    @hardmath, they did not give any definition of tangent line, I just used my understanding of what a tangent line is, I don't know if there are different definitions/meanings of tangent line. Anyway I am using seventh edition of James Steward, page 119 at the bottom – mamotebang Nov 04 '19 at 12:58

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Given a curve $C$ and a point $P$ on $C$, what does it mean to say that a line $L$ is tangent to $C$ at $P$?

Well, the first requirement is that $L$ passes through $P$. But, there are many different lines that pass through $P$. So, for instance, if $C$ is the graph of $y=|x|$ and if $P=(0,0)$ then the line $y=0$ passes through $P$, and the line $x=0$ passes through $P$, and the line $y=x$ passes through $P$, and the line $y=.01x$ passes through $P$.

What's so special about tangent lines?

The additional special property of the tangent line is not just that it passes through $P$, but that at points of the curve $C$ nearby $P$ the tangent line $L$ is a very good approximation of $C$. Intuitively, what this means is that if you took a microscope and looked at the region around $P$ through that microscope, the curve $C$ and the line $L$ would look very much like each other near $P$, but perhaps $C$ curves a bit away from $L$ somewhat. But then you have to repeat this intuition with more and more powerful microscopes. If you take a very powerful microscope, and looked at the region around $P$ through that microscope, the curve $C$ and the line $L$ would look very very much like each other near $P$, but perhaps $C$ curves a very little bit away from $L$.

Now let's apply this intuition to the case of $y=|x|$ and $P=(0,0)$. No matter how powerful of a microscope you take, when you look at the region around $P$ the graph of $y=|x|$ and the line $y=0$ look nothing like each other, other than the fact that they both contain $P$. So no, the line $y=0$ is not a tangent line to $y=|x|$, and in fact there does not exist any line is tangent to $y=|x|$ at $P$.

Of course, all these notions of very very close seem rather intuitive. But that's the point of calculus: it makes those kind of notions logically precise, using the concept of a limit. And when you make the definition of tangent line precise, for example as in the answer of @Griboullis, then you will also see that $y=|x|$ has no tangent line at $(0,0)$.

Lee Mosher
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The linear function $g(x) = \alpha x + \beta$ is tangent to $f$ at point $a$ if \begin{equation} \lim_{x\to a}\frac{f(x)-g(x)}{x-a} = 0 \end{equation} If this is true, it implies that $f(a) = g(a)$ and \begin{equation} \frac{f(x) - f(a)}{x -a} = \frac{f(x)-g(x)}{x-a} + \frac{g(x)-g(a)}{x-a} = \frac{f(x)-g(x)}{x-a} + \alpha \longrightarrow_{x\to a} \alpha \end{equation} Hence $f$ is differentiable and its derivative at point $a$ is $\alpha$

Gribouillis
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  • So, you do agree with me that y-axis is tangent to |x| at 0? because if we let f(x)=|x| and g(x) be the y axis, then g(0)=f(0) and $\alpha $=0? But $|x|$ is not differentiable at 0, so what does this mean? – mamotebang Nov 04 '19 at 12:55
  • You say '$g(x)$ be the y axis' but I'm only saying $g(x)=\alpha x + \beta$. The y-axis cannot be represented this way. There is no line tangent to the graph of $|x|$ at $0$. – Gribouillis Nov 04 '19 at 13:03
  • Sorry, I mean the x-axis – mamotebang Nov 04 '19 at 13:06
  • The $x$ axis is the graph of the linear function $g(x) = 0$. As you can see, it is not tangent to $f$ according to the definition I wrote above because $\frac{|x| - 0}{x - 0}$ doesn't tend to $0$ when $x \to 0$ – Gribouillis Nov 04 '19 at 13:07
  • @ Gribouillis, I see! So this means that it is wrong to say that a line that touches a curve once at some point it tangent to that curve? okaayy – mamotebang Nov 04 '19 at 13:15
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    Exactly, in fact the distance between the two curves at point $x$ must be much smaller than $x-a$. – Gribouillis Nov 04 '19 at 13:18
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An answer at the intuitive level :

Suppose that there is a moving object following the path determined by a curve.

Suppose that at a certain point P of this curve, the object leaves the curve and continues its way, by pure inertia.

The object would move along a straight line.

Now, this straight line is the tangent of the curve at point P just in case these 2 conditions are both fulfilled:

(1) this line is different from the original curve

(2) and this line is the same, in whatever sense the object is moving ( be it from the right to the left, or from the left to the right).

Now, suppose an object leaved the |x| curve at (0,0) : this object would not follow the same line in case it would come from the left and in case in would come from the right.

Note : The " uniqueness condition " ( condition 2) is analogous to the uniqueness condition in the definition of " the " limit of a function f as x approaches a given value; in order to be allowed to talk about " the" limit of f (when x tends to this value) , the so-called limit has to be the same as x approches this value from the right and from the left .