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Suppose two functions satisfy f(a)=g(a)=x, and for some positive integer n their nth derivatives at a are different. Also, if \[\lim_{x\to a}\frac{f(a)g(x)-f(a)-g(a)f(x)+f(a)}{g(x)-f(x)}=4,\] then the value of k is

1. 0
2. 4
3. 2
4. 1

Correct answer: 4

Solution

The limit expression simplifies to a form that reveals the behavior of the functions around the point 'a'. Given that the nth derivatives of f and g at 'a' are different, the limit evaluates to 4, indicating that the functions' growth rates differ significantly, leading to the conclusion that k must equal 4.

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