Question

Let $y=f(x)$ be a function and let $P=(a,f(a))$ and $Q(a+h,f(a+h))$ be two points close to each other on the given graph of this function.
I. $\underset{h\to 0}{\lim }\frac{f(a+h)-f(a)}{h}=\underset{Q\to P}{\lim }\frac{QR}{PR}$
II. $f^{\prime }(a)=\tan \Psi$

Which of the following is/are true?

• A. Both I and II are true
• B. Only I is true
• C. Only II is true
• D. Both I and II are false

Answer 1

Answer: (A)

We know that, $f^{\prime }(a)=\underset{h\to 0}{\lim }\frac{f(a+h)-f(a)}{h}$
From the $△PQR$, it is clear that the ratio whose limit we are taking is precisely equal to $\tan (QPR)$ which is the slope of the chord $PQ$. In the limiting process, as $h$ tends to 0 , the point $Q$ tends to $P$ and we have
$\underset{h\to 0}{\lim }\frac{f(a+h)-f(a)}{h}=\underset{Q\to P}{\lim }\frac{QR}{PR}$
This is equivalent to the fact that the chord $PQ$ tends to the tangent at $P$ of the curve $y=f(x)$. Thus, the limit turns out to be equal to the slope of the tangent. Hence,
$f^{\prime }(a)=\tan \psi .$