Question

Use continuity to evaluate the limit:

`\lim_(x \to 2)\arctan((x^2-4)/(3x^2-6x) )`

Answer 1

Find the limit...

`\lim_(x \to 2) tan^(-1)((x^2-4)/(3x^2-6x) )`

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`\Longrightarrow \lim_(x \to 2) tan^(-1)((x^2-4)/(3x^2-6x) )`

Factor the top and bottom of the expression.

`\Longrightarrow \lim_(x \to 2) tan^(-1)(((x+2)(x-2))/(3x(x-2)) )`

Cancel the common factor "(x-2)."

`\Longrightarrow \lim_(x \to 2) tan^(-1)((x+2)/(3x) )`

Now plug in the limit.

`\Longrightarrow \lim_(x \to 2) tan^(-1)((2+2)/(3(2)) ) \Longrightarrow \lim_(x \to 2) tan^(-1)((4)/(6) )=\boxed{tan^(-1)((2)/(3) )}`

`\lim_(x \to 2) tan^(-1)((x^2-4)/(3x^2-6x) )=\boxed{\boxed{tan^(-1)((2)/(3) )}} \therefore Sol.`

Thus, the limit is solved.