# indeterminate forms pdf

%PDF-1.5 %���� ��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��:$4#�� Lecture 7 : Indeterminate Forms Recall that we calculated the following limit using geometry in Calculus 1: lim x!0 sinx x = 1: De nition An indeterminate form of the type 0 0 is a limit of a quotient where both numerator and denominator approach 0. 1.2 Other Indeterminate Forms Indeterminate Forms Indeterminate Forms • The most basic indeterminate form is 0 0. %���� /Filter /FlateDecode endstream endobj 523 0 obj <>/Metadata 92 0 R/Pages 520 0 R/StructTreeRoot 108 0 R/Type/Catalog>> endobj 524 0 obj <>/MediaBox[0 0 612 792]/Parent 520 0 R/Resources<>/Font<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI]/XObject<>>>/Rotate 0/StructParents 0/Tabs/S/Type/Page>> endobj 525 0 obj <>stream h�bbdb����v�d��L�fN0��fO���"���S �~+Xd x��Y�n�F}�W�O��j��K�>$@��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��?����_* Determinate Limit Forms: Assuming that the functions involved in the limit are defined: 1. �d|���@��h&Y��[=�d.X$�v���j$v=� �� ��"�o��0�Vfk�Ia0� &�����0�mgjFrU;X�H�o�abd� �0�����@� lC� 2m�"��:�j��fe�X�cI�Ҍ@| � �?+ /Length 1995 not an indeterminate form of the type ∞ ∞, so L’Hospital’s Rule cannot be used.] %%EOF h�Ė�n�H�_�/g.�}>I�HN2��4ɮb�f$��Y��0���oU6>;s� ∞ ∞0 00 1∞ ∞ −∞ Remember that ∞ is not a real, honest number, but a shorthand for a limiting process. 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�! 4. The limit form frf results in a limit of 0. >> The limit forms a rf, for any number a, and bounded rf result in a limit of 0. So, nothing that we’ve got in our bag of tricks will work with these two limits. The limit form ff results in a limit of f. 3. This is where the subject of this section comes into play. 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 … 3 0 obj << to exist! This first is a 0/0 indeterminate form, but we can’t factor this one. %PDF-1.4 522 0 obj <> endobj �� 596 0 obj <>stream endstream endobj startxref unpredictable limit forms are called indeterminate. The limit form af, for 11a �HC#݀�_�u��@d\�J�[�(�?�Xj��ځ�v�\��xm�쇂�%v8V�o. [1ex] Condensed: The “form” 0 0 is indeterminate. 2. 0 III. The second is an $${\infty }/{\infty }\;$$ indeterminate form, but we can’t just factor an $${x^2}$$out of the numerator. ; ��Oف�,f;��U�;E.�.H.9#\� w��w)��Ѕ��r�iBI�����!��,���6���L�.i��ޣ>>��@ҝ�dתK|���'H��:��q���K'��$�?�*id�R�x�(���ӐL�5q��S��8B��1�� �]��s"���/@JQ���q�-����ϗ�kڛ� �g�k-s�_a��>p�O%FB@���Xe These indeterminate forms have many types that all require di erent techniques that will be broken down in the sections that follow. stream • It is indeterminate because, if lim x→a f(x) = lim x→a g(x) = 0, then lim x→a f(x) g(x) might equal any number or even fail to exist! A5��]E���J��HF��"ZK�2��'^p�#��E;N��ܝ$�R�;顡 �}4. I5D&�*JY�is,�$q���/_oTeaBdK�7pY�Bi��js�e�����v� 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