Mastering Limits: A Step-By-Step Guide For Evaluating Limits With Expressions Involving X

lim x->∞ pow top < pow bottom

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If we have a limit of the form:

lim x->∞ pow top < pow bottom, where pow top and pow bottom are some expressions involving x that tend to infinity as x approaches infinity, then we can use the following approach: Step 1: Divide both the numerator and the denominator by the highest power of x that is present in the expression. Step 2: Apply the limit as x approaches infinity. Step 3: If the resulting limit is of the form 0/0, ∞/∞, or ∞/-∞, use L'Hospital's rule to simplify the expression. Step 4: Apply the limit again, using the simplified expression. Step 5: If the limit exists and is finite, then that is the value of the original limit. If the limit does not exist, then we say that the limit diverges. Let's illustrate this approach with an example: Example: Find the limit as x approaches infinity of (x^3 + 4x^2 + 2)/(2x^3 - x^2 + 3). Step 1: Divide both numerator and denominator by the highest power of x, which is x^3. We get (1 + 4/x + 2/x^3) / (2 - 1/x + 3/x^3). Step 2: Apply the limit as x approaches infinity. As x approaches infinity, all terms with 1/x or 1/x^3 tend towards zero, leaving us with: 1/2 Hence, the limit as x approaches infinity of (x^3 + 4x^2 + 2)/(2x^3 - x^2 + 3) is 1/2.

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