How to Remove Ambiguity in Infinite Limits

Step-by-step Guide to Remove Ambiguity in Infinite Limits

Here is a step-by-step guide to removing ambiguity in infinite limits:

Step 1: Understanding the Ambiguity of Infinity

Firstly, grasp that the expressions involving infinity (\(∞\)) are not definite numbers but represent the idea of unbounded increase or decrease. Ambiguous forms like \(0×∞\) and \(\frac{∞}{∞}\) require a strategy that transforms them into a solvable structure, typically the \(\frac{0}{0}\) indeterminate form.

Step 2: Establishing General Principles for Infinity

Assume that \(lim_{x→c​}f(x)=±∞\) and \(lim_{x→c​}g(x)=L\), where \(L\) is a real number and not zero. Here, we have the following conditions:

• For \(L>0\), \(lim_{x→c}​[f(x)×g(x)]=±∞\) and \(lim_{x→c}​[\frac{f(x)}{g(x)}]=±∞\).
• For \(L<0\), \(lim_{x→c}​[f(x)×g(x)]=∓∞\) and \(lim_{x→c}​[\frac{f(x)}{g(x)}]=∓∞\).
• Addition or subtraction with \(±∞\) retains the infinity.

If both functions approach positive infinity, then the product and sum also tend to be positive infinity.

Step 3: Resolving the \(0×∞\) Indeterminate Form

When L is zero, we face the \(0×∞\) ambiguity. To address this, we aim to rewrite the expression to create a \(\frac{0}{0}\) form. Here’s how:

Transform \(lim_{x→c}​[f(x)×g(x)]\) into \(lim_{x→c​}\)\([\frac{g(x)}{(\frac{1}{f(x)}​)}]\). Since \(\frac{real \ number}{∞}=0\), this transformation can yield a \(\frac{0}{0}\) scenario, which is more manageable.

Step 4: Addressing the \(\frac{∞}{∞}\) Indeterminate Form

When confronting the \(\frac{∞}{∞}\) form, we apply a similar tactic by inverting both the numerator and the denominator, transforming \(lim_{x→c}​[f(x)×g(x)]\) into \(lim_{x→c​}\)\([\frac{\frac{1}{g(x)}}{(\frac{1}{f(x)})}]\), thus aiming for a \(\frac{0}{0}\) condition.

Step 5: Simplification to Escape Infinity Ambiguity

Simplify the expressions where possible:

• Factorize polynomials to cancel common terms.
• Divide each term in a rational function by the highest power of \(x\) in the denominator to normalize the limit expression.

Step 6: Applying L’Hôpital’s Rule

Once a \(\frac{0}{0}\) indeterminate form is achieved through the steps above, L’Hôpital’s Rule can be applied. This involves taking the derivative of the numerator and the derivative of the denominator and then re-evaluating the limit.

Step 7: Continuous Evaluation and Transformation

The process might need to be iterated: simplification, applying L’Hôpital’s Rule, or further transformation to expose a \(\frac{0}{0}\) form may need to be performed multiple times.

Final Word:

By methodically transforming indeterminate forms involving infinity into the more tractable \(\frac{0}{0}\) form and employing strategic calculus tools, the veil of ambiguity is lifted, allowing for the precise determination of limits at infinity. This mastery over the ethereal concept of infinity is a testament to the logical structure and elegance of mathematical principles, even when they stretch into the boundless.