Question

limit X tends to 3
x² – 3x²÷ x² – 5x + 6​

Answer 1

$\rm \longrightarrow\large \lim \limits_{x \rightarrow3}{ \dfrac{{x}^{2} - 3x}{ {x}^{2} - 5x + 6 } } \\ \\ \\ \rm \longrightarrow\large \lim \limits_{x \rightarrow3}{ \dfrac{x({x} - 3)}{ {x}(x- 2) - 3(x - 2) } }\\ \\ \\ \rm \longrightarrow\large \lim \limits_{x \rightarrow3}{ \dfrac{x({x} - 3)}{ (x- 2) (x - 3) } }\\ \\ \\ \rm \longrightarrow\large \lim \limits_{x \rightarrow3}{ \dfrac{x}{ (x- 2) } }\\ \\ \\ \rm \longrightarrow\large { \dfrac{3}{ (3- 2) } }\\ \\ \\ \rm \longrightarrow\large { \dfrac{3}{ 1 } }\\ \\ \\ \rm \longrightarrow\large3$

Answer : 3

Additional Information :-

In such type of question where we get 0/0 form , we can apply L'hospital rule. In L'hospital rule we differentiate denominator and numerator seperately. To differentiate we must know basic formulae :-

➝ d(e^x)/dx = e^x

➝ d(x^n)/dx = n x^(n-1)

➝ d(ln x)/dx = 1/x

➝ d(sin x)/dx = cos x

➝ d(cos x)/dx = - sin x

➝ d(tan x)/dx = sec² x

➝ d(sec x)/dx = tan x * sec x

➝ d(cot x)/dx = - cosec²x

➝ d(cosec x)/dx = - cosec x * cot x