document.write( "Question 1163182: Given the following function, determine if the function is continuous at x = -2
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\n" ); document.write( "\n" ); document.write( "f(x) = { (X^2-x-6)/(x+2), if x<-2
\n" ); document.write( "f(x) = { -5 , if -2≤x<3
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Algebra.Com's Answer #787186 by ikleyn(52832) $\"\"$ $\"About$

You can put this solution on YOUR website!
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document.write( "In vicinity of the point x= -2,  \r\n" );
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document.write( "    x^2 - x - 6 = (x-3)*(x+2),\r\n" );
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document.write( "so, after dividing by (x+2)\r\n" );
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document.write( "    f(x) =  = x-3  at  x < -2.\r\n" );
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document.write( "So, the limit of f(x) at x---> -2 from the left  is equal to -5.\r\n" );
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document.write( "From the other side,  f(-2) = -5, by the definition of the function f(x)  (second line).\r\n" );
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document.write( "Also, the limit f(x) at x ---> -2 from the right is equal to -5.\r\n" );
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document.write( "So, the function f(x) is CONTINUOUS at x= -2.\r\n" );
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document.write( "Next, at x= 3, the function f(x) is not formally defined, so we even can not discuss the question if it is continuous at this point.\r\n" );
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document.write( "It is NOT.\r\n" );
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\n" ); document.write( "\n" ); document.write( "An important post-solution note.\r
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document.write( "    The way how the function f(x) is defined in the third line, \r\n" );
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document.write( "        f(x) = { (x^2-9)/(x-3)  , if x≥3\r\n" );
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document.write( "    is a  HUGE mistake.   It is impossible to define a function in this way, since division by  0  at x= 3 is STRICTLY PROHIBITED.\r\n" );
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document.write( "    The correct form of this third line should be THIS\r\n" );
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document.write( "        f(x) = { (x^2-9)/(x-3)  , if x > 3.\r\n" );
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document.write( "    At x = 3, the function f(x) is UNDEFINED.\r\n" );
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