Answer to Question #246054 in Calculus for Annie

Question

Question No#08

Find the limit of each rational function as 𝒙 → ±∞. 𝑓(𝑥) = 3𝑥 + 7

 𝑥2 − 2

𝑓(𝑥)= 10𝑥5 +𝑥4 +31. 𝑥6

Question No#10

Let

ƒ(x) = 𝑥1/(1−x) .

Make tables of values of ƒ at values of x that approach x = 1 from above and below. Does ƒ(x)

appear to have a limit as x approaches 1? If so, what is it? If not, why not?

 Question No#11

Find the limits using 𝐥𝐢𝐦 𝐬𝐢𝐧 𝜽 = 𝟏 𝒙→𝟎 𝜽

a) lim 6𝑥2(cot 𝑥)(csc 2𝑥) 𝑥→0

𝑏) lim sin 3𝑦 cot 5𝑦 𝑦→0 𝑦 cot4𝑦

Answer 1

Question No#08: The first two limits are both equal to zero:

$\lim\limits_{x\, \to \,±\infin} \cfrac{3x+7}{x^2-2}=\lim\limits_{x\, \to \,±\infin} \cfrac{3x+7}{x^2-2} \Bigg( \dfrac{\frac{1}{x^2}}{\frac{1}{x^2}} \Bigg) = \\ =\lim\limits_{x\, \to \,±\infin} \cfrac{\frac{3}{x}+\frac{7}{x^2}}{1-\frac{2}{x^2}}= \cfrac{0+0}{1-0}=\cfrac{0}{1}=0$

$\lim\limits_{x\, \to \,±\infin} \cfrac{10x^5+x^4 +31}{x^6} \\ =\lim\limits_{x\, \to \,±\infin} \cfrac{10x^5+x^4 +31}{x^6} \Bigg( \dfrac{\frac{1}{x^6}}{\frac{1}{x^6}} \Bigg) \\ =\lim\limits_{x\, \to \,±\infin} \cfrac{\frac{10}{x}+\frac{1}{x^2}+\frac{31}{x^6}}{1}= \cfrac{0+0+0}{1}=0$

Question No#10

Let ƒ(x) = 𝑥1/(1−x) . Make tables of values of ƒ at values of x that approach x = 1 from above and below.

Does ƒ(x) appear to have a limit as x approaches 1? Yes, there is a limit.

If so, what is it? The limit for $f(x)=\cfrac{x}{1-x}$ can be expressed as:$\lim\limits_{x\, \to \,1^+} f(x)=+\infin \text{ and} \lim\limits_{x\, \to \,1^-} f(x)=-\infin$

Question No#11

Find the limits using $\lim\limits_{\theta\, \to \,0} \cfrac{\sin\theta}{\theta}=1$, after substitution we find

a) $\lim\limits_{x\, \to \,0} 6x^2(\cot x)(\csc 2x) \\ =\lim\limits_{x\, \to \,0} 6x^2(\frac{\cos x}{\sin x})(\frac{1}{2\sin x \cos x}) \\ =\lim\limits_{x\, \to \,0} (\frac{3x^2}{\sin^2 x})=3(\lim\limits_{x\, \to \,0} (\frac{x}{\sin x}))^2 \\ \therefore \lim\limits_{x\, \to \,0} 6x^2(\cot x)(\csc 2x) =3(1^2)=3$

𝑏) $\lim\limits_{y\, \to \,0} \cfrac{(\sin^3y)(\cot^ 3y)}{y\cot^ 4y} \\ =\lim\limits_{y\, \to \,0} \cfrac{\sin^3y}{y\cot y} =\lim\limits_{y\, \to \,0} \cfrac{\sin^3y \cos y}{y\sin y} \\ =\lim\limits_{y\, \to \,0} \cfrac{\sin^2y \cos y}{y} \\ =\lim\limits_{y\, \to \,0} \cfrac{\sin y }{y} \times \lim\limits_{y\, \to \,0} {\sin y \cos y} \\ =(1)(0)=0$

Reference:

• Thomas, G. B., & Finney, R. L. (1961). Calculus. Addison-Wesley Publishing Company.