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Carlos Module 2 Limit Revised_3.mw .docx
- 1. Carlos Module 2 Limit Revised_3.mw
- 7. vLUYsNiZGWkYvRl91Rj9GW28tRiw2JkZnbkYvRl91Rj9GW28 tRiw2JkZqbkYvRl91Rj9GaXRGYHBGPw== Calculating the limit at a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR 0aW5nR0koX3N5c2xpYkdGJzYqLUknbXVuZGVyR0YkNiUtSS Ntb0dGJDYuUSRsaW1GJy8lJXNpemVHUSMxNkYnLyUsbWF0 aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmdW5zZXR GJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3l tbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1 pdHNHUSV0cnVlRicvJSdhY2NlbnRHRjovJSdsc3BhY2VHUSY wLjBlbUYnLyUncnNwYWNlR1EsMC4xNjY2NjY3ZW1GJy1GI zYrLUkjbWlHRiQ2KFEieEYnRjIvJSdpdGFsaWNHRkUvJStmb3 JlZ3JvdW5kR1EsWzIwMCwwLDIwMF1GJy8lLHBsYWNlaG9s
- 10. UsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZ mFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0 LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFi bGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSY wLjBlbUYnLyUncnNwYWNlR0ZDRitGKy8lK2V4ZWN1dGFib GVHRjRGLw== JSFH JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR 0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Maple calculated the limit as x approaches 'a' from the right and it is equal to LUkmbWZyYWNHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2 V0dGluZ0dJKF9zeXNsaWJHRic2KC1JI21uR0YkNiRRIjFGJy8l LG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSVtcm93R0YkNiY tRiw2JFEiMkYnRi8tSSNtb0dGJDYtUSJ+RidGLy8lJmZlbmNlR 1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5 R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1v dmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VH USYwLjBlbUYnLyUncnNwYWNlR0ZNLUklbXN1cEdGJDYlL UkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnL0Yw USdpdGFsaWNGJy1GMzYkRjVGLy8lMXN1cGVyc2NyaXB0c2 hpZnRHUSIwRidGLy8lLmxpbmV0aGlja25lc3NHRi4vJStkZW5 vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGX28vJSli ZXZlbGxlZEdGPg== .
- 11. Maple calculated the limit as x approaches 'a' from the left and it is equal to LUkmbWZyYWNHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2 V0dGluZ0dJKF9zeXNsaWJHRic2KC1JI21uR0YkNiRRIjFGJy8l LG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSVtcm93R0YkNiY tRiw2JFEiMkYnRi8tSSNtb0dGJDYtUSJ+RidGLy8lJmZlbmNlR 1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5 R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1v dmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VH USYwLjBlbUYnLyUncnNwYWNlR0ZNLUklbXN1cEdGJDYlL UkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnL0Yw USdpdGFsaWNGJy1GMzYkRjVGLy8lMXN1cGVyc2NyaXB0c2 hpZnRHUSIwRidGLy8lLmxpbmV0aGlja25lc3NHRi4vJStkZW5 vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGX28vJSli ZXZlbGxlZEdGPg== . It is clear that left hand limit and right hand limit at 'a' are equal. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR 0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZt ZnJhY0dGJDYoLUYjNistSSVtc3VwR0YkNiUtRiw2J1EieEYnL
- 14. 0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= JSFH At x=a the denominator of the function will exactly equal to 0 because at x=a denominator of function is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR 0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWl HRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aH ZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiN EYnL0Y2USdub3JtYWxGJ0Y+LyUxc3VwZXJzY3JpcHRzaGlm dEdRIjBGJ0Y+- LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR 0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWl HRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aH ZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiN EYnL0Y2USdub3JtYWxGJ0Y+LyUxc3VwZXJzY3JpcHRzaGlm dEdRIjBGJ0Y+=0. That mean value of function at x=a is infinity or not defined.
- 15. Approaching a from the right a+0.01 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR 0aW5nR0koX3N5c2xpYkdGJzYnLUknbXVuZGVyR0YkNiUtSS Ntb0dGJDYvUSRsaW1GJy8lJXNpemVHUSMxNkYnLyUwZm9 udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJy8lLG1hdGh2Y XJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJnVuc2V0RicvJ SpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldH JpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR1El dHJ1ZUYnLyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4 wZW1GJy8lJ3JzcGFjZUdRLDAuMTY2NjY2N2VtRictRiM2Ky1 JI21pR0YkNilRInhGJ0YyLyUnaXRhbGljR0ZILyUrZm9yZWdy b3VuZEdRLFsyMDAsMCwyMDBdRicvJSxwbGFjZWhvbGRlck dGSEY1L0Y5USdpdGFsaWNGJy1GLzYvUS0mcmlnaHRhcnJvd ztGJ0YyRjVGOEY7Rj4vRkFGSEZCRkQvRkdGPUZJL0ZMUSw wLjI3Nzc3NzhlbUYnL0ZPRmBvLUZUNilRImFGJ0YyRldGWU ZmbkY1RmhuLUYvNi5RIitGJ0Y1RjgvRjxRJmZhbHNlRicvRj9 GaW8vRkFGaW8vRkNGaW8vRkVGaW8vRkdGaW8vRkpGaW8 vRkxRLDAuMjIyMjIyMmVtRicvRk9GYXAvJSVib2xkR0ZILy UrZXhlY3V0YWJsZUdGaW8vRjZRL0VxdWF0aW9ufkxhYmVs RicvRjlRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRmpwLyUsYWN jZW50dW5kZXJHRmlvLUZUNiNRIUYnLUkmbWZyYWNHRi Q2KC1GIzYqLUklbXN1cEdGJDYlLUkobWZlbmNlZEdGJDYm
- 21. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR 0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Here, I am just showing the behaviour of the function closer to 'a' from the right. To do that, I have chosen three points that are coming closer and closer to 'a' from the right. We can directly observe that as x comes closer and closer to 'a', the function starts to comes closer to
- 22. LUkmbWZyYWNHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2 V0dGluZ0dJKF9zeXNsaWJHRic2KC1JJW1yb3dHRiQ2JC1JI21 uR0YkNiZRIjFGJy8lJXNpemVHUSMxNkYnLyUwZm9udF9zd HlsZV9uYW1lR1EpMkR+SW5wdXRGJy8lLG1hdGh2YXJpYW 50R1Enbm9ybWFsRidGOC1GLDYmLUYvNiZRIjJGJ0YyRjVG OC1JI21vR0YkNi9RIn5GJ0YyRjVGOC8lJmZlbmNlR1EmZmFs c2VGJy8lKnNlcGFyYXRvckdGRi8lKXN0cmV0Y2h5R0ZGLyU qc3ltbWV0cmljR0ZGLyUobGFyZ2VvcEdGRi8lLm1vdmFibGVs aW1pdHNHRkYvJSdhY2NlbnRHRkYvJSdsc3BhY2VHUSYwLj BlbUYnLyUncnNwYWNlR0ZVLUklbXN1cEdGJDYlLUkjbWlH RiQ2J1EiYUYnRjIvJSdpdGFsaWNHUSV0cnVlRidGNS9GOVE naXRhbGljRictRiw2J0Y9RjJGaW5GNUZcby8lMXN1cGVyc2N yaXB0c2hpZnRHUSIwRidGOC8lLmxpbmV0aGlja25lc3NHRjEv JStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdG Z28vJSliZXZlbGxlZEdGRg== . Since the terms other than LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR 0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW5HRiQ2JlEiMkYnLy Ulc2l6ZUdRIzE2RicvJTBmb250X3N0eWxlX25hbWVHUSkyRH 5JbnB1dEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21 vR0YkNi9RIn5GJ0YvRjJGNS8lJmZlbmNlR1EmZmFsc2VGJy8l KnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV 0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHN HRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyU ncnNwYWNlR0ZNLUklbXN1cEdGJDYlLUkjbWlHRiQ2J1EiYU YnRi8vJSdpdGFsaWNHUSV0cnVlRidGMi9GNlEnaXRhbGljRic tRiM2JEYrRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjU = in the denominator starts tending to 0 as we are moving closer to 'a' from the right. JSFH
- 32. # The following code is autogenerated by the Explore command. Do not edit. # Explore:-Runtime:-Update( plot((-a^2+x^2)/(-a^4+x^4),x = -50 .. 50,labels = [x, ""],discont = [showremovable]), ':-Plot0', [':- a'], ':-sliders'=["Slidera"], ':-labels'=["Labela"],':-isplot'=true, ':-handlers'=[a=handlertab[a]]); # # End of autogenerated code. # 1. The slider used below is for taking the values of 'a' dynamically. 2. Here, the slider is limited from -10 to 10 but can be adjusted as needed. 3. The open dot is showing the position where function does not exist.
- 33. Calculating the limit at 317200 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR 0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
- 35. HNiJGKEYpRikhIiI= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR 0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Here I am calculating the right-hand limit of 'Pi'. We can find this limit by factorizing the denominator, After that ,solving the function to a simpler form and then simply by putting the value x as 'Pi'. JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR 0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZt ZnJhY0dGJDYoLUYjNistSSVtc3VwR0YkNiUtRiw2J1EieEYnL yUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJTBmb25 0X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnLyUsbWF0aHZhc mlhbnRHUSdpdGFsaWNGJy1GIzYoLUkjbW5HRiQ2JlEiMkYn RjpGQC9GRFEnbm9ybWFsRicvJSVib2xkR0Y/LyUrZXhlY3V0 YWJsZUdRJmZhbHNlRicvRkFRL0VxdWF0aW9ufkxhYmVsRic vRkRRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRlYvJTFzdXBlcn NjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2L1EoJm1pbnVz
- 44. NlNiIvJSxzdXBlcnNjcmlwdEdRJmZhbHNlNiIvJStmb3JlZ3Jvd W5kR1EoWzAsMCwwXTYiLyUrYmFja2dyb3VuZEdRLlsyNTU sMjU1LDI1NV02Ii8lJ29wYXF1ZUdRJmZhbHNlNiIvJStleGVjd XRhYmxlR1EmZmFsc2U2Ii8lKXJlYWRvbmx5R1EmZmFsc2U2 Ii8lKWNvbXBvc2VkR1EmZmFsc2U2Ii8lKmNvbnZlcnRlZEdRJ mZhbHNlNiIvJStpbXNlbGVjdGVkR1EmZmFsc2U2Ii8lLHBsY WNlaG9sZGVyR1EmZmFsc2U2Ii8lNnNlbGVjdGlvbi1wbGFjZ WhvbGRlckdRJmZhbHNlNiIvJSxtYXRodmFyaWFudEdRJ2l0Y WxpYzYiUSE2Ii0lKV9WSVNJQkxFRzYjIiIiLSUlUk9PVEc2Jy 0lKUJPVU5EU19YRzYjJCIkKyMhIiItJSlCT1VORFNfWUc2Iy QiIyEqISIiLSUtQk9VTkRTX1dJRFRIRzYjJCIlK0UhIiItJS5CT1 VORFNfSEVJR0hURzYjJCIlP0EhIiItJSlDSElMRFJFTkc2Ii0lK 0FOTk9UQVRJT05HNictJSlCT1VORFNfWEc2IyQiIiEhIiItJSlC T1VORFNfWUc2IyQiIiEhIiItJS1CT1VORFNfV0lEVEhHNiMkI iUrUyEiIi0lLkJPVU5EU19IRUlHSFRHNiMkIiUrUyEiIi0lKUNI SUxEUkVORzYiRzYi LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGlu Z0dJKF9zeXNsaWJHRic2KFEiYUYnLyUlYm9sZEdRJXRydW VGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJStleGVjdXRhYmxlR0Y xLyUsbWF0aHZhcmlhbnRHUSVib2xkRicvJStmb250d2VpZ2h0 R0Y2 # # The following code is autogenerated by the Explore command. Do not edit. # Explore:-Runtime:-Update( plot((-a^2+x^2)/(-a^4+x^4),x = - 2*Pi .. 2*Pi,resolution = 1600,labels = [x, ""],discont = [showremovable]), ':-Plot1', [':-a'], ':-sliders'=["Slidera1"], ':- labels'=["Labela1"],':-isplot'=true, ':- handlers'=[a=handlertab[a]]); # # End of autogenerated code. #
- 45. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR 0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= We can confirm that limit value of function as x tend towards LUklbXN1cEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR 0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVElJnBpO0Y nLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1 Enbm9ybWFsRictSSVtcm93R0YkNiQtRiw2I1EhRidGMi8lMXN 1cGVyc2NyaXB0c2hpZnRHUSIwRic= is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR 0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1JI 21uR0YkNiRRIjFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFs RictRiM2KS1JI21vR0YkNi1RIn5GJ0YyLyUmZmVuY2VHUSZ mYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlH Rj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW9 2YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdR JjAuMGVtRicvJSdyc3BhY2VHRkwtSSVtc3VwR0YkNiUtSSNta UdGJDYmUSUmcGk7RicvJSdpdGFsaWNHRj0vJTBmb250X3N 0eWxlX25hbWVHUSkyRH5JbnB1dEYnRjItRiM2JC1GLzYlUSI yRidGWEYyRjIvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLU
- 46. Y4Ni1RIitGJ0YyRjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjI yMmVtRicvRk5GYW8tRlA2JS1GUzYlUSJhRicvRldRJXRydW VGJy9GM1EnaXRhbGljRictRiM2Ji1GLzYkRmluRjIvJStleGVjd XRhYmxlR0Y9L0ZZUSVUZXh0RidGMkZqbkZgcEZicEYyLyU ubGluZXRoaWNrbmVzc0dGMS8lK2Rlbm9tYWxpZ25HUSdjZ W50ZXJGJy8lKW51bWFsaWduR0ZocC8lKWJldmVsbGVkR0Y 9LUY4Ni1RIi5GJ0YyRjtGPkZARkJGREZGRkhGSkZNRjI= For simplifying what I am explaining just keep the value of a=0 and check the function values closer to 'Pi', you can clearly observe that the function value is closer to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR 0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1JI 21uR0YkNiRRIjFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFs RictRiM2Jy1JI21vR0YkNi1RIn5GJ0YyLyUmZmVuY2VHUSZ mYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlH Rj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW9 2YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdR JjAuMGVtRicvJSdyc3BhY2VHRkwtSSVtc3VwR0YkNiUtSSNta UdGJDYmUSUmcGk7RicvJSdpdGFsaWNHRj0vJTBmb250X3N 0eWxlX25hbWVHUSkyRH5JbnB1dEYnRjItRiM2JC1GLzYlUSI yRidGWEYyRjIvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLU Y4Ni1RIitGJ0YyRjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjI yMmVtRicvRk5GYW8tRlA2JS1GLzYkRlxvRjItRiM2JC1GLzY kRmluRjJGMkZqbkYyLyUubGluZXRoaWNrbmVzc0dGMS8lK2 Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zfc C8lKWJldmVsbGVkR0Y9LUZTNiNRIUYnRjI= = KiQpSSNQaUclKnByb3RlY3RlZEciIiMhIiI=
- 48. EmbGFiZWxGJ0Zfby1GQTYxUSIpRidGL0YyRjhGO0ZERkZG SEZKRkxGTkZQRlJGVEZXRi9GMkY4RjtGRA== JCImSywiISIm . Similarly we can observe the limit of function by choosing the different values of 'a'. JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR 0aW5nR0koX3N5c2xpYkdGJzYoLUknbXVuZGVyR0YkNiUtSS Ntb0dGJDYvUSRsaW1GJy8lJXNpemVHUSMxNkYnLyUwZm9 udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJy8lLG1hdGh2Y XJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJnVuc2V0RicvJ SpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldH JpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR1El dHJ1ZUYnLyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4 wZW1GJy8lJ3JzcGFjZUdRLDAuMTY2NjY2N2VtRictRiM2Ky1 JI21pR0YkNilRInhGJ0YyLyUnaXRhbGljR0ZILyUrZm9yZWdy b3VuZEdRLFsyMDAsMCwyMDBdRicvJSxwbGFjZWhvbGRlck dGSEY1L0Y5USdpdGFsaWNGJy1GLzYvUS0mcmlnaHRhcnJvd ztGJ0YyRjVGOEY7Rj4vRkFGSEZCRkQvRkdGPUZJL0ZMUSw wLjI3Nzc3NzhlbUYnL0ZPRmBvLUZUNiVRJyYjOTYwO0YnL 0ZYUSZmYWxzZUYnRjgtRi82LVEoJm1pbnVzO0YnRjgvRjxG Zm8vRj9GZm8vRkFGZm8vRkNGZm8vRkVGZm8vRkdGZm8v RkpGZm8vRkxRLDAuMjIyMjIyMmVtRicvRk9GYnAvJSVib2x kR0ZILyUrZXhlY3V0YWJsZUdGZm8vRjZRL0VxdWF0aW9uf kxhYmVsRicvRjlRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRltxLy
- 50. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR 0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR 0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR 0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Here, I am calculating the left-hand limit of 'Pi'. We can find this limit by factorizing the denominator after that solving the function to simpler form and then simply putting the value x as 'Pi'. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR
- 51. 0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR 0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Calculating the limit at 342210236 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR 0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR 0aW5nR0koX3N5c2xpYkdGJzYnLUknbXVuZGVyR0YkNiUtSS Ntb0dGJDYvUSRsaW1GJy8lJXNpemVHUSMxNkYnLyUwZm9 udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJy8lLG1hdGh2Y XJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJnVuc2V0RicvJ SpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldH JpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR1El dHJ1ZUYnLyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4 wZW1GJy8lJ3JzcGFjZUdRLDAuMTY2NjY2N2VtRictRiM2Ji1J I21pR0YkNilRInhGJ0YyLyUnaXRhbGljR0ZILyUrZm9yZWdyb 3VuZEdRLFsyMDAsMCwyMDBdRicvJSxwbGFjZWhvbGRlckd
- 52. GSEY1L0Y5USdpdGFsaWNGJy1GLzYvUS0mcmlnaHRhcnJvdz tGJ0YyRjVGOEY7Rj4vRkFGSEZCRkQvRkdGPUZJL0ZMUSw wLjI3Nzc3NzhlbUYnL0ZPRmBvLUZUNiZRKWluZmluaXR5Ri cvRlhRJmZhbHNlRidGNUY4RjgvJSxhY2NlbnR1bmRlckdGZm 8tSSZtZnJhY0dGJDYoLUYjNiUtSSNtbkdGJDYmUSIxRidGMk Y1RjhGNUY4LUYjNictSSVtc3VwR0YkNiUtRlQ2J0ZWRjJGV 0Y1RmhuLUYjNiUtRl9wNiZRIjJGJ0YyRjVGOEY1RjgvJTFzd XBlcnNjcmlwdHNoaWZ0R1EiMEYnLUYvNi9RIitGJ0YyRjVG OC9GPEZmby9GP0Zmby9GQUZmby9GQ0Zmby9GRUZmby9G R0Zmby9GSkZmby9GTFEsMC4yMjIyMjIyZW1GJy9GT0Zcci1 GZXA2JS1GVDYnUSJhRidGMkZXRjVGaG5GaXBGXnFGNU Y4LyUubGluZXRoaWNrbmVzc0dGYXAvJStkZW5vbWFsaWdu R1EnY2VudGVyRicvJSludW1hbGlnbkdGZ3IvJSliZXZlbGxlZE dGZm8tRlQ2I1EhRicvJStleGVjdXRhYmxlR0Zmb0Y4 = IiIh LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR 0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Maple calculated the limit of function at 342210236 and it is equal to 0. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR
- 54. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR 0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Maple calculated the limit of function at -342210236 and it is equal to 0. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR 0aW5nR0koX3N5c2xpYkdGJzYmLUkmbWZyYWNHRiQ2KC1 GIzYnLUklbXN1cEdGJDYlLUkjbWlHRiQ2J1EieEYnLyUlc2l6 ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJTBmb250X3N0 eWxlX25hbWVHUSkyRH5JbnB1dEYnLyUsbWF0aHZhcmlhbn RHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JlEiMkYnRjdGP S9GQVEnbm9ybWFsRidGSS8lMXN1cGVyc2NyaXB0c2hpZnR HUSIwRictSSNtb0dGJDYvUSgmbWludXM7RidGN0Y9RkkvJS ZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRlQvJSlzdH JldGNoeUdGVC8lKnN5bW1ldHJpY0dGVC8lKGxhcmdlb3BHRl QvJS5tb3ZhYmxlbGltaXRzR0ZULyUnYWNjZW50R0ZULyUnb HNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGXW8t RjE2JS1GNDYnUSJhRidGN0Y6Rj1GQEZDRkstRjQ2I1EhRidG SS1GIzYmLUYxNiVGMy1GIzYkLUZGNiZRIjRGJ0Y3Rj1GSU ZJRktGTi1GMTYlRmJvRlxwRktGSS8lLmxpbmV0aGlja25lc3N HUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1 hbGlnbkdGaHAvJSliZXZlbGxlZEdGVEZlby8lK2V4ZWN1dGFi bGVHRlRGSQ== = KiYsJiokKUkiYUc2IiIiIyIiIiEiIiokKUkieEdGJ0YoRilGKUYpL CYqJClGJiIiJUYpRioqJClGLUYxRilGKUYq
- 63. =Ig== SSJhRzYi # # The following code is autogenerated by the Explore command. Do not edit. # Explore:-Runtime:-Update( plot((-a^2+x^2)/(-a^4+x^4),x = - infinity .. infinity,labels = [x, ""]), ':-Plot2', [':-a'], ':- sliders'=["Slidera2"], ':-labels'=["Labela2"],':-isplot'=true, ':- handlers'=[a=handlertab[a]]); # # End of autogenerated code. # LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR 0aW5nR0koX3N5c2xpYkdGJzZjby1JI21pR0YkNipRJXBsb3RG Jy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUlc2l6ZU dRIzE0RicvJSVib2xkR1EmZmFsc2VGJy8lJ2l0YWxpY0dRJXR ydWVGJy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStle GVjdXRhYmxlR0Y3LyUsbWF0aHZhcmlhbnRHUSdpdGFsaWN GJy1JI21vR0YkNjJRIihGJ0YvRjJGNUY7Rj4vRkFRJ25vcm1hb EYnLyUmZmVuY2VHRjovJSpzZXBhcmF0b3JHRjcvJSlzdHJld GNoeUdGOi8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjcvJ S5tb3ZhYmxlbGltaXRzR0Y3LyUnYWNjZW50R0Y3LyUnbHN wYWNlR1EsMC4xNjY2NjY3ZW1GJy8lJ3JzcGFjZUdGWS1JK G1hY3Rpb25HRiQ2JS1JKm12ZXJiYXRpbUdGJDYjUV9wKiYs JiokKUkiYUc2IiIiIyIiIiEiIiokKUkieEc2IiIiIyIiIiIiIiIiIiwmKiQp SSJhRzYiIiIlIiIiISIiKiQpSSJ4RzYiIiIlIiIiIiIiISIiRicvJSthY3Rp b250eXBlR1E7bWFwbGVzb2Z0LmNvbTpsYWJlbChMMjIxNSl
- 66. 1GLDYqUSJ5RidGL0YyRjVGOEY7Rj5GQEZjckZjb0ZdcC1GL DYqUShkaXNjb250RidGL0YyRjVGOEY7Rj5GQEZdcEZkcEZd cEZqcS1GLDYqUS5zaG93cmVtb3ZhYmxlRidGL0YyRjVGOEY 7Rj5GQEZjci1GRDYyUSIpRidGL0YyRjVGO0Y+RkdGSUZLR k1GT0ZRRlNGVUZXRlpGL0YyRjVGO0Y+Rkc= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR 0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Here, the graph is explaining that limit value at -infinity is equal to limit value at +infinity. As the function starts tending either toward -infinity or infinity, the function value starts tending toward 0. JSFH JSFH JSFH
- 67. JSFH Conclusion: Wonderful experience of learning here, learn many things like how to analyze graph for calculating limits, calculating limit at the different point of functions and how to use slider even where to use the slider as well. Got perfect knowledge of working with infinity. The most difficult thing for me was to work with the graph of maple because there was two variable 'x' and 'a' and I want to put the range -infinite to infinite to the slider for the values of 'a' but it was creating error. So, at last, I have to limit the values of 'a' to -finite to finite(example -10 to 10) values. JSFH JSFH JSFH