4.7 — / 10 Actionthri...

The key feature for Section 4.7 is , which simplifies the calculation of limits for indeterminate quotients by using derivatives.

. If the result is still indeterminate, you can apply the rule again. Example Visualization The following graph illustrates how two functions, , both approaching zero at a point

, can have a determined limit for their ratio based on their slopes (derivatives) at that point. ✅ Result 4.7 / 10 ActionThri...

∞∞the fraction with numerator infinity and denominator infinity end-fraction , the rule can be applied. : Take the derivative of the top function ( ) and the derivative of the bottom function ( ) independently. Do not use the Quotient Rule . Re-evaluate the Limit : Find the limit of the new fraction f′(x)g′(x)f prime of x over g prime of x end-fraction

∞∞the fraction with numerator infinity and denominator infinity end-fraction Feature Overview: L'Hôpital's Rule The key feature for Section 4

limx→af(x)g(x)=limx→af′(x)g′(x)limit over x right arrow a of f of x over g of x end-fraction equals limit over x right arrow a of f prime of x over g prime of x end-fraction provided the limit on the right exists (or is ±∞plus or minus infinity Step-by-Step Application

limx→af(x)=±∞ and limx→ag(x)=±∞limit over x right arrow a of f of x equals plus or minus infinity and limit over x right arrow a of g of x equals plus or minus infinity Do not use the Quotient Rule

: First, evaluate the limit directly. If it yields 000 over 0 end-fraction