Find the slant asymptote of {{MATH_46}} {{CMD_47}} We are looking to see if there is a line {{MATH_48}} such that {{MATH_49}} First, let's consider the limit as {{MATH_50}} approaches positive infinity. We will imagine that we have such a line {{MATH_51}} and attempt to find the correct values for {{MATH_52}} and {{MATH_53}}. Let's look again at our limit. We are assuming: {{MATH_54}} We know that {{MATH_55}} is continuous everywhere except at {{MATH_56}} and {{MATH_57}} is continuous everywhere, so we can apply our limit laws away from {{MATH_58}}. We're looking at large values of {{MATH_59}}, so this is no problem. We use the fact that the sum of the limits is the limit of the sums. {{ENV_60}} We are assuming these two limits are equal. Dividing by {{MATH_61}} on the right hand side makes the limit equal to {{MATH_62}}: {{ENV_63}} To find the value of {{MATH_64}}, then, we can divide the left hand side by {{MATH_65}} and evaluate the limit. We see the following. {{ENV_66}} So {{MATH_67}}. We now know that {{MATH_68}} for some value of {{MATH_69}}. To find the {{MATH_70}}-intercept {{MATH_71}}, we use a similar method. Notice that {{MATH_72}} so if we subtract {{MATH_73}} from the right hand side, we are left with just {{MATH_74}}. Since the two sides are equal, subtracting {{MATH_75}} from the left hand side and evaluating the limit will give us the value for {{MATH_76}}. We write the following. {{ENV_77}} By this method, we have determined that {{MATH_78}} In other words, {{MATH_79}} is a slant asymptote for our function {{MATH_80}}. You should check that we get the same slant asymptote {{MATH_81}} when we take the limit to negative infinity as well. {{CMD_82}} We can confirm our results by looking at the graph of {{MATH_83}} and {{MATH_84}}: {{MATH_85}} {{CMD_86}} {{CMD_87}} {{CMD_88}}