009A Sample Final 3, Problem 1

Find each of the following limits if it exists. If you think the limit does not exist provide a reason.

(a)  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lim _{x\rightarrow 0}{\frac {\sin(5x)}{1-{\sqrt {1-x}}}}}

(b)  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lim _{x\rightarrow 8}f(x),}   given that  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lim _{x\rightarrow 8}{\frac {xf(x)}{3}}=-2}

(c)  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lim _{x\rightarrow -\infty }{\frac {\sqrt {9x^{6}-x}}{3x^{3}+4x}}}

Foundations:
1. If  $\lim _{x\rightarrow a}g(x)\neq 0,$  we have
$\lim _{x\rightarrow a}{\frac {f(x)}{g(x)}}={\frac {\displaystyle {\lim _{x\rightarrow a}f(x)}}{\displaystyle {\lim _{x\rightarrow a}g(x)}}}.$
2.  $\lim _{x\rightarrow 0}{\frac {\sin x}{x}}=1$

Solution:

(a)

Step 1:
We begin by noticing that we plug in  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 x=0}   into
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{\sin(5x)}{1-\sqrt{1-x}},}
we get   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{0}{0}.}

Step 2:
Now, we multiply the numerator and denominator by the conjugate of the denominator.
Hence, we have
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 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{\bigg(\lim_{x\rightarrow 0} \frac{\sin(5x)}{x}\bigg) \lim_{x\rightarrow 0}(1+\sqrt{1-x})}\\ &&\\ & = & \displaystyle{\bigg(5\lim_{x\rightarrow 0} \frac{\sin(5x)}{5x}\bigg) (2)}\\ &&\\ & = & \displaystyle{5(1)(2)}\\ &&\\ & = & \displaystyle{10.} \end{array}}

(b)

Step 1:
Since  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} 3 =3\ne 0,}
we have
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{-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}}

Step 2:
If we multiply both sides of the last equation by  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 3,}   we get
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 -6=\lim_{x\rightarrow 8} xf(x).}
Now, using properties of limits, we have
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{-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}}

Step 3:
Solving for  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)}   in the last equation,
we get
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}.}

(c)

Step 1:
First, we write
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(1-\frac{1}{9x^5})}}{3x^3+4x}}\\ &&\\ & = & \displaystyle{\lim_{x\rightarrow -\infty} \frac{3|x^3|\sqrt{(1-\frac{1}{9x^5})}}{3x^3+4x}}\\ &&\\ & = & \displaystyle{\lim_{x\rightarrow -\infty} \frac{3(-x^3)\sqrt{(1-\frac{1}{9x^5})}}{3x^3(1+\frac{4}{3x^2})}} \end{array}}

Step 2:
Now, we have
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{3(-x^3)\sqrt{(1-\frac{1}{9x^5})}}{3x^3(1+\frac{4}{3x^2})}}\\ &&\\ & = & \displaystyle{\lim_{x\rightarrow -\infty} \frac{-\sqrt{(1-\frac{1}{9x^5})}}{1+\frac{4}{3x^2}}}\\ &&\\ & = & \displaystyle{\frac{-\sqrt{1}}{1}}\\ &&\\ & = & \displaystyle{-1.}\\ \end{array}}

Final Answer:
(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}
(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}}
(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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