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(-6+x)/(3+sqrt(3+x))
Limit of the function/ (-6+x)/(3+sqrt(3+x))

Limit of the function (-6+x)/(3+sqrt(3+x))

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     /    -6 + x   \
 lim |-------------|
x->6+|      _______|
     \3 + \/ 3 + x /
$$\lim_{x \to 6^+}\left(\frac{x - 6}{\sqrt{x + 3} + 3}\right)$$ 
Limit((-6 + x)/(3 + sqrt(3 + x)), x, 6)
Lopital's rule
There is no sense to apply Lopital's rule to this function since there is no indeterminateness of 0/0 or oo/oo type
The graph
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Rapid solution [src] 
0
$$0$$
Expand and simplify
One‐sided limits [src] 
     /    -6 + x   \
 lim |-------------|
x->6+|      _______|
     \3 + \/ 3 + x /
$$\lim_{x \to 6^+}\left(\frac{x - 6}{\sqrt{x + 3} + 3}\right)$$ 
0
$$0$$ 
= 2.04767629454407e-33
     /    -6 + x   \
 lim |-------------|
x->6-|      _______|
     \3 + \/ 3 + x /
$$\lim_{x \to 6^-}\left(\frac{x - 6}{\sqrt{x + 3} + 3}\right)$$ 
0
$$0$$ 
= -1.08543171911275e-33
= -1.08543171911275e-33
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 6^-}\left(\frac{x - 6}{\sqrt{x + 3} + 3}\right) = 0$$
More at x→6 from the left
$$\lim_{x \to 6^+}\left(\frac{x - 6}{\sqrt{x + 3} + 3}\right) = 0$$
$$\lim_{x \to \infty}\left(\frac{x - 6}{\sqrt{x + 3} + 3}\right) = \infty$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{x - 6}{\sqrt{x + 3} + 3}\right) = - \frac{6}{\sqrt{3} + 3}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x - 6}{\sqrt{x + 3} + 3}\right) = - \frac{6}{\sqrt{3} + 3}$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{x - 6}{\sqrt{x + 3} + 3}\right) = -1$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x - 6}{\sqrt{x + 3} + 3}\right) = -1$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x - 6}{\sqrt{x + 3} + 3}\right) = \infty i$$
More at x→-oo
Numerical answer [src] 
2.04767629454407e-33
2.04767629454407e-33
The graph
Limit of the function (-6+x)/(3+sqrt(3+x))
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