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Maths-

General

Easy

Question

$L subscript left parenthesis t rightwards arrow 2 right parenthesis left parenthesis vertical line x minus 2 vertical line right parenthesis divided by left parenthesis x minus 2 right parenthesis$

-1
1
0
2

hint Hint:

The given function is modulus function so we will calculate Left hand limit (LHL) and Right Hand limit (RHL).

The correct answer is: -1

In this question we have to find
$limit as t minus greater than 2 of fraction numerator open vertical bar x minus 2 close vertical bar over denominator x minus 2 end fraction$ . We are given modulus function. So, we have to check limit in the neighborhood of $2$
Step1: Checking limit at $t minus greater than 2 to the power of plus$ . That is Right Hand Limit
$limit as t minus greater than 2 to the power of plus of fraction numerator open vertical bar x minus 2 close vertical bar over denominator x minus 2 end fraction$
Since, $x greater than 2 comma space open vertical bar x minus 2 close vertical bar equals x minus 2$
$limit as t minus greater than 2 to the power of plus of fraction numerator x minus 2 over denominator x minus 2 end fraction$
=> $limit as t minus greater than 2 to the power of plus of fraction numerator x minus 2 over denominator x minus 2 end fraction equals 1$
Step2: Checking Left hand limit
$limit as t minus greater than 2 to the power of minus of fraction numerator open vertical bar x minus 2 close vertical bar over denominator x minus 2 end fraction$
Since, $x less than 2 comma space open vertical bar x minus 2 close vertical bar equals space minus left parenthesis x minus 2 right parenthesis$
$limit as t minus greater than 2 to the power of minus of fraction numerator negative left parenthesis x minus 2 right parenthesis over denominator x minus 2 end fraction$
=> $limit as t minus greater than 2 to the power of minus of fraction numerator negative left parenthesis x minus 2 right parenthesis over denominator x minus 2 end fraction equals negative 1$
Since both left and Right hand limit are not same the function is not continuous at $x equals 2$

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