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Find whether f(x) is continuous or discontinuous at the indicated point $f(x)= x ^2+x-1\;at\;x=1.$

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Toolbox:
  • A function is said to be continuous at a point $'a'$ if the LHL = RHL.
  • (i.e) $\lim\limits_{\large x\to a^+}f(x)=\lim\limits_{\large x\to a^-}f(x)$
Step 1:
Given : $f(x)=x^2+x-1$ at $x=1$
The left hand limit :
$\lim\limits_{\large x\to 1^-}f(x)=\lim\limits_{\large h\to 0}f(1-h)$
$\qquad\qquad=\lim\limits_{\large h\to 0}(1-h)^2+(1-h)-1$
$\qquad\qquad=\lim\limits_{\large h\to 0}1+h^2-2h+1-h-1$
On applying limits we get,
$\qquad\qquad=1$
Step 2:
The right hand limit :
$\lim\limits_{\large x\to 1^+}f(x)=\lim\limits_{\large h\to 0}f(1+h)$
$\qquad\qquad=\lim\limits_{\large h\to 0}(1+h)^2+(1+h)-1$
$\qquad\qquad=\lim\limits_{\large h\to 0}1+h^2+2h+1+h-1$
On applying limits we get,
$\qquad\qquad=1$
Hence $\lim\limits_{\large x\to 1^-}f(x)=\lim\limits_{\large x\to 1^+}f(x)$
This proves the continuity of the function at $x=1$
answered Jun 26, 2013 by sreemathi.v
 

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