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Question

# If f:R→R is defined byf(x)=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩x+2x2+3x+2,ifx∈R−{−1,−2}−1,ifx=−20,ifx=−1then f is continuous on the set

A
R
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B
R{2}
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C
R{1}
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D
R{1,2}
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Solution

## The correct option is B R−{−1}Since, f(x) is continuous for every value of R except {−1,−2}. Now, we have to check that points.At x=−2, L.H.L =limn→0(−2−n)+2(−2−n)2+3(−2−n)+2=limn→0−nn2+n=−1R.H.L =limn→0(−2+n)+2(−2+n)2+3(−2+n)+2=limn→0nn2−n=−1∴LHL=RHL=f(−2)∴ f is continuous at x=−2Now, check for x=−1L.H.L =limn→0(−1−n)+2(−1−n)2+3(−1−n)+2=limn→01−nn2−n=∞R.H.L =limn→0(−1+n)+2(−1+n)2+3(−1+n)+2 =limn→01+nn2+n=∞∴LHL=RHL≠f(−1)∴ f is not continuous at x=−1Hence, the function f is continuous on the set R−{−1}

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