This is a beautiful problem from ISI Mstat 2019 PSA problem 17 based on limit of a function . We provide sequential hints so that you can try this.

Limit of a function


If \( f(a)=2, f'(a)=1 , g(a)=-1\) and \(g'(a)=2\) , then what is

 \( \lim\limits_{x\to a}\frac{(g(x)f(a) – g(a)f(x))}{(x – a)} \) ?

  • 5
  • 3
  • – 3
  • -5

Key Concepts


Algebraic manipulation

Limit form of the Derivative

Check the Answer


But try the problem first…

Answer: is 5

Source
Suggested Reading

ISI MStat 2019 PSA Problem 17

Introduction to real analysis Robert G. Bartle, Donald R., Sherbert.

Try with Hints


First hint

Try to manipulate \( \frac{(g(x)f(a) – g(a)f(x))}{(x – a)} \) so that you can use the Limit form of the Derivative . Let’s give a try .

Second Hint

\( \frac{(g(x)f(a) – g(a)f(x))}{(x – a)} \) =

\( \frac{(g(x)f(a) –g(a)f(a) +g(a)f(a) – g(a)f(x))}{(x – a)} \) =

\( f(a)\frac{g(x)-g(a)}{(x-a)} – g(a)\frac{f(x)-f(a)}{(x-a)} \) .

Now calculate the limit using Limit form of the Derivative.

Final Step

So, we have \( \lim\limits_{x\to a}\frac{(g(x)f(a) – g(a)f(x)}{(x – a)} \) =

\( \lim\limits_{x\to a} f(a)\frac{g(x)-g(a)}{(x-a)} – \lim\limits_{x\to a} g(a)\frac{f(x)-f(a)}{(x-a)} \) =

\( f(a) g'(a) – g(a)f'(a)= 2.(2)-1.(-1)=5 \).

Outstanding Statistics Program with Applications

Outstanding Statistics Program with Applications

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