I5D&�*JY�is,�$q���/_oTeaBdK�7pY�Bi��js�e�����v� 2m�"��:�j��fe`�X�c`I�Ҍ@| � �?+ %PDF-1.5 %���� �d|���@��h&Y��`[=�d.X$�v��`�j$v=� �� ��"�o��0�Vfk�Ia0� &�����0�mgjFrU;X�H�o�a`bd`� �0�����@� lC� The limit form frf results in a limit of 0. 522 0 obj <> endobj 3 0 obj << The limit form af, for 11a Lecture 7 : Indeterminate Forms Recall that we calculated the following limit using geometry in Calculus 1: lim x!0 sinx x = 1: De nition An indeterminate form of the type 0 0 is a limit of a quotient where both numerator and denominator approach 0. These indeterminate forms have many types that all require di erent techniques that will be broken down in the sections that follow. Indeterminate Form of the Type 0⋅∞ Indeterminate forms of the type 0⋅∞ can sometimes be evaluated by rewriting the product as a quotient, and then applying L’Hospital’s Rule for the indeterminate forms of … 572 0 obj <>/Filter/FlateDecode/ID[<5279F4B2F385564FAAF522E8F6B2E2AF>]/Index[522 75]/Info 521 0 R/Length 172/Prev 288589/Root 523 0 R/Size 597/Type/XRef/W[1 3 1]>>stream /Filter /FlateDecode endstream endobj 523 0 obj <>/Metadata 92 0 R/Pages 520 0 R/StructTreeRoot 108 0 R/Type/Catalog>> endobj 524 0 obj <>/MediaBox[0 0 612 792]/Parent 520 0 R/Resources<>/Font<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI]/XObject<>>>/Rotate 0/StructParents 0/Tabs/S/Type/Page>> endobj 525 0 obj <>stream to exist! Determinate Limit Forms: Assuming that the functions involved in the limit are defined: 1. 596 0 obj <>stream 4. %���� So, nothing that we’ve got in our bag of tricks will work with these two limits. >> 0 1.2 Other Indeterminate Forms Indeterminate Forms Indeterminate Forms • The most basic indeterminate form is 0 0. %%EOF �HC#݀�_�u��@d\�J�[�(�?�Xj��`ځ�v�\��xm�쇂�%v8V�o. h�Ė�n�H�_�/g.�}>I�HN2��4ɮb�f$��Y��0���oU6>;s� h�b``�c``����� }�A�����,,*��x8/v87� l;� R"S�s���f�#�O�p��`�h(�`�h�h�h��@����b�� ��l��m��2mc�e��x��6�3&>۟7�~eX�p�����aC�F��m���a�H?C4�! The limit form ff results in a limit of f. 3. /Length 1995 The second is an \({\infty }/{\infty }\;\) indeterminate form, but we can’t just factor an \({x^2}\)out of the numerator. ∞ ∞0 00 1∞ ∞ −∞ Remember that ∞ is not a real, honest number, but a shorthand for a limiting process. This is where the subject of this section comes into play. The limit forms a rf, for any number a, and bounded rf result in a limit of 0. unpredictable limit forms are called indeterminate. A5��]E���J��HF��"ZK�2`��'^p�#��E;N��ܝ$�R�;顡 �}4. III. h�bbd```b``����`v�d��L�fN0��fO���"���S �~+Xd ��b7��s]�9���u��$Bj���mT�5#���~mo�]����ݵ�2�ϛ�f�t7�f�k���������[8wВ)�/�"�����������M}r�5}�Yp[��M��n�\�w�����m��t���P�^���6�wv]z{��כ�D�9͍��v�V���#t���f����* �]J)�p�&�����Ɣ�/���%�;\���p7���".e0�x�m��;7����Qt�oK/��I����zou����K.ڷ��рp���o����M��.�T�N �D��>ê�N6�u�w��MZ$ �~�B���z���L�� x���z�ޓ� ����iB��������8G��:$4#�� endstream endobj startxref %PDF-1.4 �� not an indeterminate form of the type ∞ ∞, so L’Hospital’s Rule cannot be used.] stream ; ��Oف�,f;��U�;`E.�.H.9#\� w��w)��Ѕ��r`�iBI�����!��,���6���L�.i��ޣ>>��@ҝ�dתK|���'H��:��q���K'��$�?�*id�R�x�(���ӐL�5q��S��8B��1�� �]��s"���/@JQ���q�-����ϗ�kڛ� �g�k-s�_a��>p�O%FB@���Xe x��Y�n�F}�W�O��j��K�>$@��h�1Ї��M�Dd�!�:�����Z�+ٹ���"���3g�̬^��|�#S��QǪ��J��pC�ķ�����W+�ӟK!D͘�b���Ms�k��@֒�(��v����-�9�����d��.����o?��'?���?a�����v�h�r~s��?����_* • 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! [1ex] Condensed: The “form” 0 0 is indeterminate. This first is a 0/0 indeterminate form, but we can’t factor this one. 2.
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