If $f(2) = 4$ and $f'(2) = 1$, then $\displaystyle\lim_{x \to 2} \frac{xf (2) - 2 f (x) }{x -2}$ is equal to
- -2
- 1
- 2
- 3
The Correct Option is C
Solution and Explanation
$=\displaystyle\lim _{x \rightarrow 2} \frac{x f(2)-2 f(2)+2 f(2)-2 f(x)}{x-2}$
$=\displaystyle\lim _{x \rightarrow 2} \frac{(x-2) f(2)}{x-2}-2 \displaystyle\lim _{x \rightarrow 2} \frac{f(x)-f(2)}{x-2}$
$=f(2)-2 f'(2)=4-2 \times 1=2$
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Concepts Used:
Limits
A function's limit is a number that a function reaches when its independent variable comes to a certain value. The value (say a) to which the function f(x) approaches casually as the independent variable x approaches casually a given value "A" denoted as f(x) = A.
If limx→a- f(x) is the expected value of f when x = a, given the values of ‘f’ near x to the left of ‘a’. This value is also called the left-hand limit of ‘f’ at a.
If limx→a+ f(x) is the expected value of f when x = a, given the values of ‘f’ near x to the right of ‘a’. This value is also called the right-hand limit of f(x) at a.
If the right-hand and left-hand limits concur, then it is referred to as a common value as the limit of f(x) at x = a and denote it by lim x→a f(x).