Answer to Question #129849 in Calculus for Hetisani Sewela

Question #129849

Use L’Hopital’s rule to evaluate: ˆ

(a) limx→0 (1-cos^3x)/sin^2x

(b) limx→∞ (cos 1/x)^x

(c) limx→∞ x ln x/(x^2 - 1)

(d) limx→1 x ln x /(x^2 - 1)


1
Expert's answer
2020-08-20T17:58:32-0400

a)applying L Hopital's rule


"\\lim _{x\\to \\:0}\\frac{3\\cos ^2\\left(x\\right)\\sin \\left(x\\right)}{2sinxcosx}"


"\\lim _{x\\to \\:0}\\frac{3\\cos ^2\\left(x\\right)\\sin \\left(x\\right)}{\\sin \\left(2x\\right)}"

Again applying the rule ,

"\\lim _{x\\to \\:0}\\left(\\frac{3\\left(-\\sin \\left(2x\\right)\\sin \\left(x\\right)+\\cos ^3\\left(x\\right)\\right)}{\\cos \\left(2x\\right)\\cdot \\:2}\\right)"

Plugin the value for x=0,

"=\\frac{3\\left(-\\sin \\left(2\\cdot \\:0\\right)\\sin \\left(0\\right)+\\cos ^3\\left(0\\right)\\right)}{\\cos \\left(2\\cdot \\:0\\right)\\cdot \\:2}"

= "\\frac{3}{2}"


b) "\\lim _{x\\to \\infty \\:}\\left(\\cos ^x\\left(\\frac{1}{x}\\right)\\right)"


"\\mathrm{Apply\\:exponent\\:rule}:\\quad \\:a^x=e^{\\ln \\left(a^x\\right)}=e^{x\\cdot \\ln \\left(a\\right)}"


"\\cos ^x\\left(\\frac{1}{x}\\right)=e^{x\\ln \\left(\\cos \\left(\\frac{1}{x}\\right)\\right)}"


"=\\lim _{x\\to \\infty \\:}\\left(e^{x\\ln \\left(\\cos \\left(\\frac{1}{x}\\right)\\right)}\\right)"


"\\mathrm{Apply\\:the\\:Limit\\:Chain\\:Rule}:\\lim _{x\\to \\infty \\:}\\left(e^{\\frac{\\ln \\left(\\cos \\left(\\frac{1}{x}\\right)\\right)}{\\frac{1}{x}}}\\right)"


Applying L Hopitals's rule "\\lim _{x\\to \\infty \\:}\\left(e^{\\frac{\\frac{\\tan \\left(\\frac{1}{x}\\right)}{x^2}}{-\\frac{1}{x^2}}}\\right)"

"\\lim _{x\\to \\infty \\:}e^{\\left(-\\tan \\left(\\frac{1}{x}\\right)\\right)}"

= "\\lim _{x\\to \\infty \\:}e^{\\left(-\\tan \\left(\\frac{1}{\\infty}\\right)\\right)}"

= "\\lim _{x\\to \\infty \\:}e^{\\left(-\\tan (0)\\right)}"

"\\lim _{x\\to \\infty \\:}e^{(0)}"

=1


Hence "\\lim _{x\\to \\infty \\:}\\left(\\cos ^x\\left(\\frac{1}{x}\\right)\\right)=1"


c)"\\mathrm{Divide\\:by\\:highest\\:denominator\\:power:}\\:\\frac{\\frac{\\ln \\left(x\\right)}{x}}{1-\\frac{1}{x^2}}"

"=\\lim _{x\\to \\infty \\:}\\left(\\frac{\\frac{\\ln \\left(x\\right)}{x}}{1-\\frac{1}{x^2}}\\right)"

"=\\frac{\\lim _{x\\to \\infty \\:}\\left(\\frac{\\ln \\left(x\\right)}{x}\\right)}{\\lim _{x\\to \\infty \\:}\\left(1-\\frac{1}{x^2}\\right)}"

"\\lim _{x\\to \\infty \\:}\\left(\\frac{\\ln \\left(x\\right)}{x}\\right)=0" and "{\\lim _{x\\to \\infty \\:}\\left(1-\\frac{1}{x^2}\\right)}=1"

="\\frac{0}{1}" = 0


d)Applying L Hopital's rule

"=\\lim _{x\\to \\:1}\\left(\\frac{\\ln \\left(x\\right)+1}{2x}\\right)"


"\\mathrm{Plug\\:in\\:the\\:value}\\:x=1"

"=\\frac{\\ln \\left(1\\right)+1}{2\\cdot \\:1}"

"=\\frac{1}{2}"




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Comments

Assignment Expert
22.09.20, 17:32

Dear sire, please use the panel for submitting new questions.

sire
22.09.20, 15:14

1) ∫(1+√y)^1/4 /√y dy 2) I=∫sec^2(πlnx)/3x dx, x>1 3) I=∫(tan(4ϴ)^-1/4 sec^2(4ϴ) dϴ 4) I=∫cos(2+ln√x)/x dx, x>1

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