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Limit Calculator
Calculate limits step by step
This free calculator will try to find the limit (two-sided or one-sided, including left and right) of the given function at the given point (including infinity), with steps shown.
Solution
Your input: find $$$\lim_{x \to 3^\mathtt{\text{-}}} \frac{\sqrt{9 - x^{2}}}{\sqrt{x} + \sqrt{3 - x} - \sqrt{3}}$$$
Since we have an indeterminate form of type $$$\frac{0}{0}$$$, we can apply the l'Hopital's rule:
$${\color{red}{\lim_{x \to 3^\mathtt{\text{-}}} \frac{\sqrt{9 - x^{2}}}{\sqrt{x} + \sqrt{3 - x} - \sqrt{3}}}} = {\color{red}{\lim_{x \to 3^\mathtt{\text{-}}} \frac{\frac{d}{dx}\left(\sqrt{9 - x^{2}}\right)}{\frac{d}{dx}\left(\sqrt{x} + \sqrt{3 - x} - \sqrt{3}\right)}}}$$
For steps, see derivative calculator.
$${\color{red}{\lim_{x \to 3^\mathtt{\text{-}}} \frac{\frac{d}{dx}\left(\sqrt{9 - x^{2}}\right)}{\frac{d}{dx}\left(\sqrt{x} + \sqrt{3 - x} - \sqrt{3}\right)}}} = {\color{red}{\lim_{x \to 3^\mathtt{\text{-}}}\left(- \frac{x}{\sqrt{9 - x^{2}} \left(- \frac{1}{2 \sqrt{3 - x}} + \frac{1}{2 \sqrt{x}}\right)}\right)}}$$
Simplify:
$${\color{red}{\lim_{x \to 3^\mathtt{\text{-}}}\left(- \frac{x}{\sqrt{9 - x^{2}} \left(- \frac{1}{2 \sqrt{3 - x}} + \frac{1}{2 \sqrt{x}}\right)}\right)}} = {\color{red}{\lim_{x \to 3^\mathtt{\text{-}}} \frac{2 x^{\frac{3}{2}} \sqrt{3 - x}}{\sqrt{9 - x^{2}} \left(\sqrt{x} - \sqrt{3 - x}\right)}}}$$
Apply the constant multiple rule $$$\lim_{x \to 3^\mathtt{\text{-}}} c f{\left(x \right)} = c \lim_{x \to 3^\mathtt{\text{-}}} f{\left(x \right)}$$$ with $$$c=2$$$ and $$$f{\left(x \right)} = \frac{x^{\frac{3}{2}} \sqrt{3 - x}}{\sqrt{9 - x^{2}} \left(\sqrt{x} - \sqrt{3 - x}\right)}$$$:
$${\color{red}{\lim_{x \to 3^\mathtt{\text{-}}} \frac{2 x^{\frac{3}{2}} \sqrt{3 - x}}{\sqrt{9 - x^{2}} \left(\sqrt{x} - \sqrt{3 - x}\right)}}} = {\color{red}{\left(2 \lim_{x \to 3^\mathtt{\text{-}}} \frac{x^{\frac{3}{2}} \sqrt{3 - x}}{\sqrt{9 - x^{2}} \left(\sqrt{x} - \sqrt{3 - x}\right)}\right)}}$$
The limit of a product/quotient is the product/quotient of limits:
$$2 {\color{red}{\lim_{x \to 3^\mathtt{\text{-}}} \frac{x^{\frac{3}{2}} \sqrt{3 - x}}{\sqrt{9 - x^{2}} \left(\sqrt{x} - \sqrt{3 - x}\right)}}} = 2 {\color{red}{\lim_{x \to 3^\mathtt{\text{-}}} \frac{x^{\frac{3}{2}}}{\sqrt{x} - \sqrt{3 - x}} \lim_{x \to 3^\mathtt{\text{-}}} \frac{\sqrt{3 - x}}{\sqrt{9 - x^{2}}}}}$$
The limit of a quotient is the quotient of limits:
$$2 \lim_{x \to 3^\mathtt{\text{-}}} \frac{\sqrt{3 - x}}{\sqrt{9 - x^{2}}} {\color{red}{\lim_{x \to 3^\mathtt{\text{-}}} \frac{x^{\frac{3}{2}}}{\sqrt{x} - \sqrt{3 - x}}}} = 2 \lim_{x \to 3^\mathtt{\text{-}}} \frac{\sqrt{3 - x}}{\sqrt{9 - x^{2}}} {\color{red}{\frac{\lim_{x \to 3^\mathtt{\text{-}}} x^{\frac{3}{2}}}{\lim_{x \to 3^\mathtt{\text{-}}}\left(\sqrt{x} - \sqrt{3 - x}\right)}}}$$
Substitute the variable with the value:
$$\frac{2 \lim_{x \to 3^\mathtt{\text{-}}} \frac{\sqrt{3 - x}}{\sqrt{9 - x^{2}}} {\color{red}{\lim_{x \to 3^\mathtt{\text{-}}} x^{\frac{3}{2}}}}}{\lim_{x \to 3^\mathtt{\text{-}}}\left(\sqrt{x} - \sqrt{3 - x}\right)} = \frac{2 \lim_{x \to 3^\mathtt{\text{-}}} \frac{\sqrt{3 - x}}{\sqrt{9 - x^{2}}} {\color{red}{\left(3 \sqrt{3}\right)}}}{\lim_{x \to 3^\mathtt{\text{-}}}\left(\sqrt{x} - \sqrt{3 - x}\right)}$$
Substitute the variable with the value:
$$6 \sqrt{3} \lim_{x \to 3^\mathtt{\text{-}}} \frac{\sqrt{3 - x}}{\sqrt{9 - x^{2}}} {\color{red}{\lim_{x \to 3^\mathtt{\text{-}}}\left(\sqrt{x} - \sqrt{3 - x}\right)}}^{-1} = 6 \sqrt{3} \lim_{x \to 3^\mathtt{\text{-}}} \frac{\sqrt{3 - x}}{\sqrt{9 - x^{2}}} {\color{red}{\left(\sqrt{3}\right)}}^{-1}$$
Move the limit under the power:
$$6 {\color{red}{\lim_{x \to 3^\mathtt{\text{-}}} \frac{\sqrt{3 - x}}{\sqrt{9 - x^{2}}}}} = 6 {\color{red}{\sqrt{\lim_{x \to 3^\mathtt{\text{-}}} \frac{3 - x}{9 - x^{2}}}}}$$
Factor:
$$6 \sqrt{{\color{red}{\lim_{x \to 3^\mathtt{\text{-}}} \frac{3 - x}{9 - x^{2}}}}} = 6 \sqrt{{\color{red}{\lim_{x \to 3^\mathtt{\text{-}}} \frac{3 - x}{\left(-1\right) \left(x - 3\right) \left(x + 3\right)}}}}$$
Cancel the common term:
$$6 \sqrt{{\color{red}{\lim_{x \to 3^\mathtt{\text{-}}} \frac{3 - x}{\left(-1\right) \left(x - 3\right) \left(x + 3\right)}}}} = 6 \sqrt{{\color{red}{\lim_{x \to 3^\mathtt{\text{-}}} \frac{1}{x + 3}}}}$$
Substitute the variable with the value:
$$6 \sqrt{{\color{red}{\lim_{x \to 3^\mathtt{\text{-}}} \frac{1}{x + 3}}}} = 6 \sqrt{{\color{red}{\left(\frac{1}{6}\right)}}}$$
Therefore,
$$\lim_{x \to 3^\mathtt{\text{-}}} \frac{\sqrt{9 - x^{2}}}{\sqrt{x} + \sqrt{3 - x} - \sqrt{3}} = \sqrt{6}$$
Answer: $$$\lim_{x \to 3^\mathtt{\text{-}}} \frac{\sqrt{9 - x^{2}}}{\sqrt{x} + \sqrt{3 - x} - \sqrt{3}}=\sqrt{6}$$$