What is the limit as x approaches 1 of (x^{3a}-3ax+3a-1)/((x-1)^2) ?

The limit as x approaches 1 of (x^{3a}-3ax+3a-1)/((x-1)^2) is (3a(3a-1))/2

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limit as x approaches 1 of (x^{3a}-3ax+3a-1)/((x-1)^2)

x \to 1 lim (\frac{x ^{3 a} - 3 a x + 3 a - 1}{( x - 1 ) ^{2}})

\frac{3 a ( 3 a - 1 )}{2}

Solution steps

x \to 1 lim (\frac{x ^{3 a} - 3 a x + 3 a - 1}{( x - 1 ) ^{2}})

Apply L'Hopital's Rule

= x \to 1 lim (\frac{3 a x ^{3 a - 1} - 3 a}{2 ( x - 1 )})

Apply L'Hopital's Rule

= x \to 1 lim (\frac{3 a x ^{3 a - 2} ( 3 a - 1 )}{2})

Plug in the value

x = 1

= \frac{3 a \cdot 1 ^{3 a - 2} ( 3 a - 1 )}{2}

Simplify

= \frac{3 a ( 3 a - 1 )}{2}

What is the limit as x approaches 1 of (x^{3a}-3ax+3a-1)/((x-1)^2) ?
The limit as x approaches 1 of (x^{3a}-3ax+3a-1)/((x-1)^2) is (3a(3a-1))/2