• It is indeterminate because, if lim x→a f(x) = lim x→a g(x) = 0, then lim x→a f(x) g(x) might equal any number or even fail to exist! 18.01 Calculus Jason Starr Due by 2:00pm sharp Fall 2005 Friday, Dec. 9, 2005 Solution to (b) By plugging in x = π/2, we see that the limit becomes ”1∞”, which IS an indeterminate form. ~�-,1F��U�����1|��G�7 �B�#xxq^�j9��V� >> endobj 1 of 10 Instructions: (1) Read the problem and answer choices carefully (2) Work the problems on paper as needed (3) Pick the answer (4) Go back to review the … �8J�l���͊:�'i;��,���%���x��0�yVٛh! Lecture 7 : Indeterminate Forms Recall that we calculated the following limit using geometry in Calculus 1: lim x!0 sinx x = 1: De nition An indeterminate form of the type 0 0 is a limit of a quotient where both numerator and denominator approach 0. "���u�s�U�����U�~���/��r��>[G����w���Nj7/^�n�K$�q������H��b]��:J3O�����Lg9ł�s��\o��`�0���*n�w�e1����A]1�Ĥ�H�8�Vg [X�m���qf���l��$&yCL1kle�w=�)�*�q�R���E�=If�ӑc��# ���.kMҠ�+�[Q����Z1�s4sDX�>�Ւz���9�z���C̢"��?�N4�q#0,#s���SGR[f���neyWL�d!�Z���>�]�R�u�W�:��%���Wg�>��k��1�.gg�?�� �K�;�)�hr�yBX�gܜM6��O�V�_U=ؑ\���y����1��0�����q5 �N�ٞ���}g�3����wn**T�j`�$��L(?1��&�5���&I�� ��̃h%��V���5T��SҪ�ᮙ��*��WZ��6d9d���rǦ�ҍ�A\��:M6dM6� ��!�{�{"����'=p�>��`4��V�&�ZT� V:+���Q�)�(�aN�yC��t�07�&hR�kIpo�V�0��k�,>Ib�d��B>#;}���g3�� �QN��f 1 of 10 Instructions: (1) Read the problem and answer choices carefully (2) Work the problems on paper as needed (3) Pick the answer (4) Go back to review the … �� G��M���^O�z^�ե����Js^������a��-����ܫr�>�? x��[�G��#��/��q�GP�0ŐHج����/|�{��_OU��3=o7�S��癞��:u���f>0�/�y�bw��م��V~=۽����V���7p5Ke�p�h��rmg'��4��^����|6Lz1>��̌1� >s�H�o'1k%�ϧ�� �Q�Ӟ��% |��V‹Q�Rf�2�Ϳ�u�ם�3��ovG��~�3��j��_����i�9��-9��E�};㬷�S|��{*?�Cn���F{��^���sǤ�v~*a�׹y�#�g�3�x���v�X;���g\Y8Z�Jq!��p��V�G?NZ�gǓ�K3 ��r3�h �hm� ߝ�_o㡂3��ga���+-������ �! ∞into 0 1/∞ or into ∞ 1/0, for example one can write lim x→∞xe −x as lim x→∞x/e xor as lim x→∞e −x/(1/x). Sa�X$\֏���:,�d�z�O����:�6��㭔� ��͂5��-�ȏe�t��p�� ^tT��6^])�* /Filter /FlateDecode /Filter /FlateDecode 1.2 Other Indeterminate Forms Indeterminate Forms Indeterminate Forms • The most basic indeterminate form is 0 0. Such a problem is known as an indeterminate form 0 0. To see that the exponent forms are indeterminate note that %���� It is “indeterminate” because we just can’t conclude, on the basis of f(x) and g(x) what the limit is, or even if it exists. endstream /Type /Page /Font << /F23 4 0 R /F15 5 0 R /F27 6 0 R /F30 7 0 R /F24 8 0 R /F26 9 0 R /F29 10 0 R /F28 11 0 R /F25 12 0 R >> f4� >> endobj /MediaBox [0 0 612 792] ��~a�9��-�jz�1 But for this particular example, we can transform it as 1 sinx 1 x = x sinx xsinx; which has the standard indeterminate form \0 0". �. /Resources 1 0 R ∞ ∞0 00 1∞ ∞ −∞ Remember that ∞ is not a real, honest number, but a shorthand for a limiting process. [1ex] Condensed: The “form” 0 0 is indeterminate. Worksheet 6: (4.4-5.2) Solutions Indeterminate Forms and L’Hospital’s Rule 1. �׋��t�@���aK>N=���L��hL�S�b��F��I�����ѷ-�7��r>���#R���l�5��d���r|\�c������t��nj�K���>��}��8!Mx`x����+��c܉y���#Ր��ӲǗ6���x��0]� �9FO2z3q�™:�%���TSg�^� u��QJ !.�^�#�Q����v��;��*��,6�/�+��mo�� To see that the exponent forms are indeterminate note that '������)׆�*�����������>�����&q�;���Nw���f�1��g-�_D]�n�"�1�k�^���R~Dcb:S�x��� stream ni�����Ыόq'���/�\`��{t);)��;f"IfT*�p�Pfut xڽXKs�6��W�7j:D ,��顏��L/m}KrPe�֌�,���I��S����}|�����-x�댷ן�go�$B0��,�W��Y� B.E. /ProcSet [ /PDF /Text ] 5 0 obj /Length 1854 endobj /Parent 13 0 R We can force a common denominator: 1 1 e u 1 u = u 1 + e u u(1 e u): As u !0+, the right-hand-side is now a \0=0" form and can be treated using l’H^opital’s rule. This limit has the indeterminate form \11 ", which we haven’t men-tioned. stream Understanding their indeterminate forms is crucial. Usually, it is best to find a common factor or find a common denominator to convert it into a form where L’Hopital’s rule can be used. By Ketul Vaidya(15BEMEG061) Prashant Ranade(15BEMEG062) Aakash Singh(15BEMEG063) Aditya(15BEMEG064) Swapnil Bodke(15BEMEG065) F.Y. <> ~E3���Dvԅ�˯�'1�{R��a6P�ji-�,^O'�m������,3�py��U!�l6W4�I�����%(g✓"���nvj�6�G��_Y���Fo�(�Thg�V�ՇO����l�1�)5d���À5{/3{��£>�o:ޞXa&���if�N� ��t�'�b?�&O��oC=��z�WUBY=!�. >> You have also encountered another indeterminate form, namely ∞ ∞. Examples with detailed solutions and exercises that solves limits questions related to indeterminate forms such as : Theorem A second version of L'Hopital's rule allows us to replace the limit problem ߽)8S�E�YA�ظ-ޝ�VbX�;��0�A�IG�dm�t������pQ" NK�jE��ί&��j6��7��f5�E�MD����)P��xY |���H8� ?+&��8�y`����l�y&�h: ���⤙�i&M�s�b�� �[�ժ��S�~cT�~�����x�x�cΗ��z��vx�f�]�* �g���{^Y��#��{;@h�*@Z-���5�����F��{��1�tOC�� %�쏢 ∞ ∞0 00 1∞ ∞ −∞ Remember that ∞ is not a real, honest number, but a shorthand for a limiting process. 1 0 obj << There is in fact no general way to evaluate limits of such forms. Before we can use l’Hopital’s rule we must bring the expression into an indeterminate form of either of the two types: 0 0 or 1 1. 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