009A Sample Midterm 3, Problem 1

Find the following limits:

a) If 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 3} \bigg(\frac{f(x)}{2x}+1\bigg)=2,} find $\lim _{x\rightarrow 3}f(x).$

b) Find 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 0} \frac{\tan(4x)}{\sin(6x)}. }

c) Evaluate 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 \infty} \frac{-2x^3-2x+3}{3x^3+3x^2-3}. }

Foundations:
1. Linearity rules of limits
2. lim sin(x)/x

Solution:

(a)

Step 1:
First, 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 3} \bigg(\frac{f(x)}{2x}+1\bigg)}\\ &&\\ & = & \displaystyle{\lim_{x\rightarrow 3} \frac{f(x)}{2x}+\lim_{x\rightarrow 3} 1}\\ &&\\ & = & \displaystyle{\lim_{x\rightarrow 3} \frac{f(x)}{2x}+1.} \end{array}}
Therefore,
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 3} \frac{f(x)}{2x}=1.}

Step 2:
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 3} 2x=6\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{1} & = & \displaystyle{\lim_{x\rightarrow 3} \frac{f(x)}{2x}}\\ &&\\ & = & \displaystyle{\frac{\lim_{x\rightarrow 3} f(x)}{\lim_{x\rightarrow} 2x}}\\ &&\\ & = & \displaystyle{\frac{\lim_{x\rightarrow 3} f(x)}{6}.} \end{array}}
Multiplying both sides 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 6,} 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 3} f(x)=6.}

(b)

Step 1:

Step 2:

(c)

Step 1:

Step 2:

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 \lim_{x\rightarrow 3} f(x)=6}
(b)
(c)

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