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Question
If the following function is continuous at x = 2 then the value of k will be ______.
f(x) = `{{:(2x + 1",", if x < 2),( k",", if x = 2),(3x - 1",", if x > 2):}`
Options
2
3
5
– 1
Solution
If the following function is continuous at x = 2 then the value of k will be 5.
f(x) = `{{:(2x + 1",", if x < 2),( k",", if x = 2),(3x - 1",", if x > 2):}`
Explanation:
k = 5
∵ f(x) is continuous at x = 2
∴ LHL = RHL = f(2) ...(i)
Now LHL = `lim_(h -> 0) f(2 - h)`
= `lim_(h -> 0) 2(2 - h) + 1`
= `lim_(h -> 0) 5 - 2h`
= 5 – 0
= 5
RHL = `lim_(h -> 0) f(2 + h)`
= `lim_(h -> 0) 3(2 + h) - 1`
= `lim_(h -> 0) 5 + 3h`
= 5 + 0
= 5
And f(2) = k
Then by (i),
LHL = RHL = f(2)
`\implies` k = 5
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