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If f(x)=2/(x-3),g(x)=(x-3)/(x+4) , and h...

If `f(x)=2/(x-3),g(x)=(x-3)/(x+4)` , and `h(x)=-(2(2x+1))/(x^2+x-12)` then ` lim_(x->3)[f(x)+g(x)+h(x)]` is (a) `-2` (b) `-1` (c) `-2/7` (d) `0`

A

1

B

`oo`

C

`sqrt(2)`

D

none of these

Text Solution

AI Generated Solution

To solve the limit \( \lim_{x \to 3} [f(x) + g(x) + h(x)] \) where \( f(x) = \frac{2}{x-3} \), \( g(x) = \frac{x-3}{x+4} \), and \( h(x) = -\frac{2(2x+1)}{x^2+x-12} \), we will follow these steps: ### Step 1: Analyze the functions First, we note that \( f(x) \) has a vertical asymptote at \( x = 3 \) since the denominator becomes zero. Therefore, we need to analyze the limit carefully. ### Step 2: Simplify \( h(x) \) The function \( h(x) \) can be simplified. The denominator \( x^2 + x - 12 \) can be factored: \[ ...
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