tangents conditions

Question

what are the condition for a tangent to be exist . Is it necessary for the function to be continous.

but it is necessary to be continous for a function to be differentiable at that point .

can anybody tell me the basic concept about all this

Answer 1

A function has a tangent at $a$ if and only if it is differentiable at $a$. A function is continuous at $a$ if it is differentiable at $a$. (But note a function can be continuous at $a$ but not differentiable there.) Thus you are right that for a function to have a tangent at $a$ it must be continuous at $a$.

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In response to your comments below:

The tangent line to a function $f$ at $a$ is given by $$f(a)+(x-a)f'(a).$$ Its slope is the derviative of $f$ at $a$.

The derivative cannot be infinite since by definition, the derivative of $f$ at $a$, denoted $f'(a)$ is

$$f'(a)=\lim_{x\to a} \frac{f(x)-f(a)}{x-a}$$

provided the limit exists and is finite.

The fact that the derivative is finite does not prevent it going to infinity. That is while we cannot have $f'(a)=\infty$, we can have $\lim_{x\to a}f'(x)=\infty$. For example if $f:(0,\infty)\to\mathbb{R}$ is the function given by $f(x)=\ln x$ for all $x\in(0,\infty)$ then $f'(x)=\frac{1}{x}$ and $$\lim_{x\to 0}f'(x)=\lim_{x\to 0}\frac{1}{x}=\infty.$$

The slope of the tangent lines goes to infinity in the limit. To put it another way, the tangent lines limit to a vertical line. Note that $f'(0)$ is not even defined (and so cannot be infinite) because $f(0)$ is not defined ($0$ is not in the domain of the function $f$). In fact there is no choice of $f(0)$ for which $f$ would be differentiable at $0$.