Why every point of a function where differentiation exists has only one tangent? Can anyone help me out?
Why every point of a function where differentiation exists has only one tangent?
I know the slope at any point of any function is defined by differentiation at that point.But there may be another straight line which touches the function at that point but slope is different from the differentiation value at that point. But this does not happen(I am not considering singular point).

The value of the derivative is unique (if it exists) because it is defined as a limit, and the value of a limit is unique. Indeed, if we assume that $c$ and $b$ are distinct and 
$$\lim_{x\to a}f(x) = c, \lim_{x\to a}f(x) = b$$
Then we can pick $\epsilon = \frac{|b-c|}{2}$, and so there exists $\delta$ so that $|x-a|<\delta$ implies both $|f(x) - b| < \epsilon$ and $|f(x) - c| < \epsilon$. But then
$$|b-c| = |b-f(x)+f(x)-c| \leq|b-f(x)|+|c-f(x)|<2\epsilon=|b-c|$$ 
Which is a contradiction. So, the value of a limit (and therefore the value of a derivative, if it exists) is unique. 
