If $f: R \rightarrow R$ is defined by $f(x)=[x-3]+|x-4|$ for $x \in R$, then $\displaystyle\lim _{x \rightarrow 3} f(x)$ is equal to
- -2
- -1
- 0
- 1
The Correct Option is C
Solution and Explanation
$f(x)=[x-3]+|x-4|$
$\therefore \displaystyle\lim _{x \rightarrow 3^{-}} f(x)=\displaystyle\lim _{x \rightarrow 3^{-}} x-3+x-4$
$=\displaystyle\lim _{h \rightarrow 0} 3-h-3+3-h-4$
$=\displaystyle\lim _{h \rightarrow 0}-h+1+h$
$=-1+1+0=0$
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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).
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