Compute: {{MATH_80}} {{CMD_81}} First let's look at the form of this limit. We do this by taking the limits of both the numerator and denominator separately: {{MATH_82}} So, this limit is of the form {{CMD_83}}. This form is {{CMD_84}}determinate{{CMD_85}}, since it implies that the limit does not exist.{{CMD_86}} But, we can do better than that! As {{MATH_87}} approaches {{MATH_88}}: {{CMD_89}} {{CMD_90}} The numerator is a {{CMD_91}}{{CMD_92}}positive{{CMD_93}}{{CMD_94}}negative{{CMD_95}}{{CMD_96}} number. {{CMD_97}} The denominator is {{CMD_98}}{{CMD_99}}positive{{CMD_100}}{{CMD_101}}negative{{CMD_102}}{{CMD_103}} and is approaching zero. {{CMD_104}} This means that {{MATH_105}} {{CMD_106}} {{CMD_107}} In this example, when we say ``As {{MATH_108}} approaches {{MATH_109}}'' we don't mean that we're checking a single value of {{MATH_110}}. We mean ALL values of {{MATH_111}} that are close enough to {{MATH_112}}. Checking a single value is not enough to justify whether the numerator will be positive or negative for all {{MATH_113}} values sufficiently close to {{MATH_114}}. Constructing a sign-chart or graphing the numerator could be helpful with this determination.