Find each of the following limits if it exists. If you think the limit does not exist provide a reason.
(a) 
(b) 
given that 
(c) 
| Background Information:
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| 1. If Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lim _{x\rightarrow a}g(x)\neq 0,}
we have
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| Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lim _{x\rightarrow a}{\frac {f(x)}{g(x)}}={\frac {\displaystyle {\lim _{x\rightarrow a}f(x)}}{\displaystyle {\lim _{x\rightarrow a}g(x)}}}.}
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| 2. Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lim _{x\rightarrow 0}{\frac {\sin x}{x}}=1}
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Solution:
(a)
| Step 1:
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We begin by noticing that we plug in  into
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| Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {\sin(5x)}{1-{\sqrt {1-x}}}},}
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| we get Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {0}{0}}.}
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| Step 2:
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| Now, we multiply the numerator and denominator by the conjugate of the denominator.
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| Hence, we have
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| Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{array}{rcl}\displaystyle {\lim _{x\rightarrow 0}{\frac {\sin(5x)}{1-{\sqrt {1-x}}}}}&=&\displaystyle {\lim _{x\rightarrow 0}{\frac {\sin(5x)}{1-{\sqrt {1-x}}}}{\bigg (}{\frac {1+{\sqrt {1-x}}}{1+{\sqrt {1-x}}}}{\bigg )}}\\&&\\&=&\displaystyle {\lim _{x\rightarrow 0}{\frac {\sin(5x)(1+{\sqrt {1-x}})}{x}}}\\&&\\&=&\displaystyle {\lim _{x\rightarrow 0}{\frac {\sin(5x)}{x}}(1+{\sqrt {1-x}})}\\&&\\&=&\displaystyle {\lim _{x\rightarrow 0}{\bigg (}{\frac {\sin(5x)}{x}}{\bigg )}\lim _{x\rightarrow 0}(1+{\sqrt {1-x}})}\\&&\\&=&\displaystyle {5\lim _{x\rightarrow 0}{\bigg (}{\frac {\sin(5x)}{5x}}{\bigg )}(2)}\\&&\\&=&\displaystyle {5(1)(2)}\\&&\\&=&\displaystyle {10.}\end{array}}}
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(b)
| Step 1:
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Since 
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| we have
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| Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{array}{rcl}\displaystyle {-2}&=&\displaystyle {\lim _{x\rightarrow 8}{\bigg [}{\frac {xf(x)}{3}}{\bigg ]}}\\&&\\&=&\displaystyle {\frac {\displaystyle {\lim _{x\rightarrow 8}xf(x)}}{\displaystyle {\lim _{x\rightarrow 8}3}}}\\&&\\&=&\displaystyle {{\frac {\displaystyle {\lim _{x\rightarrow 8}xf(x)}}{3}}.}\end{array}}}
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| Step 2:
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If we multiply both sides of the last equation by  we get
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| Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle -6=\lim _{x\rightarrow 8}xf(x).}
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| Now, using properties of limits, we have
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| Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{array}{rcl}\displaystyle {-6}&=&\displaystyle {{\bigg (}\lim _{x\rightarrow 8}x{\bigg )}{\bigg (}\lim _{x\rightarrow 8}f(x){\bigg )}}\\&&\\&=&\displaystyle {8\lim _{x\rightarrow 8}f(x).}\\\end{array}}}
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| Step 3:
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| Solving for Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lim _{x\rightarrow 8}f(x)}
in the last equation,
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| we get
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Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\rightarrow 8} f(x)=-\frac{3}{4}.}
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(c)
| Step 1:
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| First, we write
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| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{\lim_{x\rightarrow -\infty} \frac{\sqrt{9x^6-x}}{3x^3+4x}} & = & \displaystyle{\lim_{x\rightarrow -\infty} \frac{\sqrt{9x^6-x}}{3x^3+4x}\cdot \frac{\big(\frac{1}{x^3}\big)}{\big(\frac{1}{x^3}\big)}}\\ &&\\ & = & \displaystyle{\lim_{x\rightarrow -\infty} \frac{\sqrt{9-\frac{1}{x^5}}}{3+\frac{4}{x^2}}.} \end{array}}
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| Step 2:
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| Now, we have
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| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{\lim_{x\rightarrow -\infty} \frac{\sqrt{9x^6-x}}{3x^3+4x}} & = & \displaystyle{\frac{\displaystyle{\lim_{x\rightarrow -\infty}} \sqrt{9-\frac{1}{x^5}}}{\displaystyle{\lim_{x\rightarrow -\infty}}\bigg(3+\frac{4}{x^2}\bigg)}}\\ &&\\ & = & \displaystyle{\frac{\sqrt{9}}{3}}\\ &&\\ & = & \displaystyle{1.} \end{array}}
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| Final Answer:
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| (a) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10}
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| (b) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{3}{4}}
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| (c) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1}
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