# Determine the value of the constant a if the function #f(x)# defined below is continuous at #x=2#. #f(x)={(ax^2+7x; x≤2),(3x^2+3a; x>2):}# ?

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If the piecewise function is continuous at

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Please observe that the graph is continuous.

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To determine the value of the constant ( a ) such that the function ( f(x) ) is continuous at ( x = 2 ), we need to ensure that the limit of the function as ( x ) approaches ( 2 ) from both sides is equal.

- First, find the expression for ( f(x) ) when ( x ) approaches ( 2 ) from the left side (denoted as ( f(2^-) )):

[ f(2^-) = a(2)^2 + 7(2) ]

- Next, find the expression for ( f(x) ) when ( x ) approaches ( 2 ) from the right side (denoted as ( f(2^+) )):

[ f(2^+) = 3(2)^2 + 3a ]

- Since the function is defined as continuous at ( x = 2 ), ( f(2^-) ) should equal ( f(2^+) ):

[ a(2)^2 + 7(2) = 3(2)^2 + 3a ]

[ 4a + 14 = 12 + 3a ]

[ 4a - 3a = 12 - 14 ]

[ a = -2 ]

So, the value of the constant ( a ) that makes the function ( f(x) ) continuous at ( x = 2 ) is ( a = -2 ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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