4.8E: Ejercicios para la Sección 4.8
- Page ID
- 116494
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)En los ejercicios 1 - 6, evaluar el límite.
1) Evaluar el límite\(\displaystyle \lim_{x→∞}\frac{e^x}{x}\).
2) Evaluar el límite\(\displaystyle \lim_{x→∞}\frac{e^x}{x^k}\).
- Responder
- \(\displaystyle \lim_{x→∞}\frac{e^x}{x^k} \quad = \quad ∞\)
3) Evaluar el límite\(\displaystyle \lim_{x→∞}\frac{\ln x}{x^k}\).
4) Evaluar el límite\(\displaystyle \lim_{x→a}\frac{x−a}{x^2−a^2}\).
- Responder
- \(\displaystyle \lim_{x→a}\frac{x−a}{x^2−a^2} \quad = \quad \frac{1}{2a}\)
5. Evaluar el límite\(\displaystyle \lim_{x→a}\frac{x−a}{x^3−a^3}\).
6. Evaluar el límite\(\displaystyle \lim_{x→a}\frac{x−a}{x^n−a^n}\).
- Responder
- \(\displaystyle \lim_{x→a}\frac{x−a}{x^n−a^n} \quad = \quad \frac{1}{na^{n−1}}\)
En los ejercicios 7 - 11, determina si puedes aplicar directamente la regla de L'Hôpital. Explique por qué o por qué no. Entonces, indica si hay alguna manera de que puedas alterar el límite para que puedas aplicar la regla de L'Hôpital.
7)\(\displaystyle \lim_{x→0^+}x^2\ln x\)
8)\(\displaystyle \lim_{x→∞}x^{1/x}\)
- Responder
- No se puede aplicar directamente; usar logaritmos
9)\(\displaystyle \lim_{x→0}x^{2/x}\)
10)\(\displaystyle \lim_{x→0}\frac{x^2}{1/x}\)
- Responder
- No se puede aplicar directamente; reescribir como\(\displaystyle \lim_{x→0}x^3\)
11)\(\displaystyle \lim_{x→∞}\frac{e^x}{x}\)
En los ejercicios 12 - 40, evaluar los límites ya sea con la regla de L'Hôpital o con métodos previamente aprendidos.
12)\(\displaystyle \lim_{x→3}\frac{x^2−9}{x−3}\)
- Responder
- \(\displaystyle \lim_{x→3}\frac{x^2−9}{x−3} \quad = \quad 6\)
13)\(\displaystyle \lim_{x→3}\frac{x^2−9}{x+3}\)
14)\(\displaystyle \lim_{x→0}\frac{(1+x)^{−2}−1}{x}\)
- Responder
- \(\displaystyle \lim_{x→0}\frac{(1+x)^{−2}−1}{x} \quad = \quad -2\)
15)\(\displaystyle \lim_{x→π/2}\frac{\cos x}{\frac{π}{2}−x}\)
16)\(\displaystyle \lim_{x→π}\frac{x−π}{\sin x}\)
- Responder
- \(\displaystyle \lim_{x→π}\frac{x−π}{\sin x} \quad = \quad -1\)
17)\(\displaystyle \lim_{x→1}\frac{x−1}{\sin x}\)
18)\(\displaystyle \lim_{x→0}\frac{(1+x)^n−1}{x}\)
- Responder
- \(\displaystyle \lim_{x→0}\frac{(1+x)^n−1}{x} \quad = \quad n\)
19)\(\displaystyle \lim_{x→0}\frac{(1+x)^n−1−nx}{x^2}\)
20)\(\displaystyle \lim_{x→0}\frac{\sin x−\tan x}{x^3}\)
- Responder
- \(\displaystyle \lim_{x→0}\frac{\sin x−\tan x}{x^3} \quad = \quad −\frac{1}{2}\)
21)\(\displaystyle \lim_{x→0}\frac{\sqrt{1+x}−\sqrt{1−x}}{x}\)
22)\(\displaystyle \lim_{x→0}\frac{e^x−x−1}{x^2}\)
- Responder
- \(\displaystyle \lim_{x→0}\frac{e^x−x−1}{x^2} \quad = \quad \frac{1}{2}\)
23)\(\displaystyle \lim_{x→0}\frac{\tan x}{\sqrt{x}}\)
24)\(\displaystyle \lim_{x→1}\frac{x-1}{\ln x}\)
- Responder
- \(\displaystyle \lim_{x→1}\frac{x-1}{\ln x} \quad = \quad 1\)
25)\(\displaystyle \lim_{x→0}\,(x+1)^{1/x}\)
26)\(\displaystyle \lim_{x→1}\frac{\sqrt{x}−\sqrt[3]{x}}{x−1}\)
- Responder
- \(\displaystyle \lim_{x→1}\frac{\sqrt{x}−\sqrt[3]{x}}{x−1} \quad = \quad \frac{1}{6}\)
27)\(\displaystyle \lim_{x→0^+}x^{2x}\)
28)\(\displaystyle \lim_{x→∞}x\sin\left(\tfrac{1}{x}\right)\)
- Responder
- \(\displaystyle \lim_{x→∞}x\sin\left(\tfrac{1}{x}\right) \quad = \quad 1\)
29)\(\displaystyle \lim_{x→0}\frac{\sin x−x}{x^2}\)
30)\(\displaystyle \lim_{x→0^+}x\ln\left(x^4\right)\)
- Responder
- \(\displaystyle \lim_{x→0^+}x\ln\left(x^4\right) \quad = \quad 0\)
31)\(\displaystyle \lim_{x→∞}(x−e^x)\)
32)\(\displaystyle \lim_{x→∞}x^2e^{−x}\)
- Responder
- \(\displaystyle \lim_{x→∞}x^2e^{−x} \quad = \quad 0\)
33)\(\displaystyle \lim_{x→0}\frac{3^x−2^x}{x}\)
34)\(\displaystyle \lim_{x→0}\frac{1+1/x}{1−1/x}\)
- Responder
- \(\displaystyle \lim_{x→0}\frac{1+1/x}{1−1/x} \quad = \quad -1\)
35)\(\displaystyle \lim_{x→π/4}(1−\tan x)\cot x\)
36)\(\displaystyle \lim_{x→∞}xe^{1/x}\)
- Responder
- \(\displaystyle \lim_{x→∞}xe^{1/x} \quad = \quad ∞\)
37)\(\displaystyle \lim_{x→0}x^{1/\cos x}\)
38)\(\displaystyle \lim_{x→0^{+} }x^{1/x}\)
- Responder
- \(\displaystyle \lim_{x→0^{+} }x^{1/x} \quad = \quad 0\)
39)\(\displaystyle \lim_{x→0}\left(1−\frac{1}{x}\right)^x\)
40)\(\displaystyle \lim_{x→∞}\left(1−\frac{1}{x}\right)^x\)
- Responder
- \(\displaystyle \lim_{x→∞}\left(1−\frac{1}{x}\right)^x \quad = \quad \frac{1}{e}\)
Para los ejercicios 41 - 50, usa una calculadora para graficar la función y estimar el valor del límite, luego usa la regla de L'Hôpital para encontrar el límite directamente.
41) [T]\(\displaystyle \lim_{x→0}\frac{e^x−1}{x}\)
42) [T]\(\displaystyle \lim_{x→0}x\sin\left(\tfrac{1}{x}\right)\)
- Responder
- \(\displaystyle \lim_{x→0}x\sin\left(\tfrac{1}{x}\right) \quad = \quad 0\)
43) [T]\(\displaystyle \lim_{x→1}\frac{x−1}{1−\cos(πx)}\)
44) [T]\(\displaystyle \lim_{x→1}\frac{e^{x−1}−1}{x−1}\)
- Responder
- \(\displaystyle \lim_{x→1}\frac{e^{x−1}−1}{x−1} \quad = \quad 1\)
45) [T]\(\displaystyle \lim_{x→1}\frac{(x−1)^2}{\ln x}\)
46) [T]\(\displaystyle \lim_{x→π}\frac{1+\cos x}{\sin x}\)
- Responder
- \(\displaystyle \lim_{x→π}\frac{1+\cos x}{\sin x} \quad = \quad 0\)
47) [T]\(\displaystyle \lim_{x→0}\left(\csc x−\frac{1}{x}\right)\)
48) [T]\(\displaystyle \lim_{x→0^+}\tan\left(x^x\right)\)
- Responder
- \(\displaystyle \lim_{x→0^+}\tan\left(x^x\right) \quad = \quad \tan 1\)
49) [T]\(\displaystyle \lim_{x→0^+}\frac{\ln x}{\sin x}\)
50) [T]\(\displaystyle \lim_{x→0}\frac{e^x−e^{−x}}{x}\)
- Responder
- \(\displaystyle \lim_{x→0}\frac{e^x−e^{−x}}{x} \quad = \quad 2\)