Limits Continuity And Differentiability Question 37
Question: Let $ f(x)= \begin{cases} 3x-4, & 0\le x\le 2 \\ 2x+\ell , & 2<x\le 9 \\ \end{cases} . $ If f is continuous at x = 2, then what is the value of $ \ell $ ?
Options:
A) 0
B) 2
C) -2
D) -1
Show Answer
Answer:
Correct Answer: C
Solution:
Given function is: $ f(x)= \begin{cases} 3x-4, & 0\le x\le 2 \\ 2x+\ell , & 2<x\le 9 \\ \end{cases} . $ and also given that f(x) is continuous at $ x=2 $ . For a function to be continuous at a point LHL = RHL = V.F. at that point. $ f(2)=2=V.F. $
$ \Rightarrow RHL:\underset{x\to 2}{\mathop{\lim }}(2x+\ell )=3(2)-4 $
$ \Rightarrow \underset{h\to 0}{\mathop{\lim }}{ 2(2+h)+\ell }=6-4 $
$ \Rightarrow 4+\ell =2,\Rightarrow \ell =-2 $
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