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- https://math.libretexts.org/Bookshelves/Calculus/Calculus_(Guichard)/04%3A_Transcendental_Functions/4.10%3A_Limits_RevisitedWhile some limits are easy to see, others take some ingenuity; in particular, the limits that define derivatives are always difficult for ratios which have both the numerator and denominator approach ...While some limits are easy to see, others take some ingenuity; in particular, the limits that define derivatives are always difficult for ratios which have both the numerator and denominator approach zero.
- https://math.libretexts.org/Bookshelves/Analysis/A_Primer_of_Real_Analysis_(Sloughter)/06%3A_Derivatives/6.05%3A_L'Hopital's_Rulelimx→a+f′(x)g′(x)=λ. whenever x∈(a,a+δ). Now, by the Generalized Mean Value Theorem, for any x and y with \(a<x<y<a+\delt...limx→a+f′(x)g′(x)=λ. whenever x∈(a,a+δ). Now, by the Generalized Mean Value Theorem, for any x and y with a<x<y<a+δ, there exists a point c∈(x,y) such that Suppose a,b∈R,f and g are differentiable on (a,b),g′(x)≠0 for all x∈(a,b), and limx→b−f′(x)g′(x)=λ.
- https://math.libretexts.org/Bookshelves/Calculus/Calculus_3e_(Apex)/06%3A_Techniques_of_Integration/6.07%3A_L'Hopital's_RuleWhile this chapter is devoted to learning techniques of integration, this section is not about integration. Rather, it is concerned with a technique of evaluating certain limits that will be useful in...While this chapter is devoted to learning techniques of integration, this section is not about integration. Rather, it is concerned with a technique of evaluating certain limits that will be useful in the following section, where integration is once more discussed. This section introduces L'Hôpital's Rule, a method of resolving limits that produce the indeterminate forms 0/0 and ∞/∞.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_I%3A_Differential_Calculus/04%3A_Applications_of_Derivatives/4.07%3A_LHopitals_RuleIn this section, we examine a powerful tool for evaluating limits. This tool, known as L’Hôpital’s rule, uses derivatives to calculate limits. With this rule, we will be able to evaluate many limits w...In this section, we examine a powerful tool for evaluating limits. This tool, known as L’Hôpital’s rule, uses derivatives to calculate limits. With this rule, we will be able to evaluate many limits we have not yet been able to determine. Instead of relying on numerical evidence to conjecture that a limit exists, we will be able to show definitively that a limit exists and to determine its exact value.
- https://math.libretexts.org/Courses/Mount_Royal_University/Calculus_for_Scientists_I/4%3A_Applications_of_Derivatives/4.8%3A_L'H%C3%B4pital's_Rulelimx→0+sinxlnx=limx→0+sinx1/lnx=limx→0+cosx−1/(x(lnx)2)=limx→0+(−x(lnx)2cosx). if f and g are differentiable functions over an int...limx→0+sinxlnx=limx→0+sinx1/lnx=limx→0+cosx−1/(x(lnx)2)=limx→0+(−x(lnx)2cosx). if f and g are differentiable functions over an interval a, except possibly at a, and limx→af(x)=0=limx→ag(x) or limx→af(x) and limx→ag(x) are infinite, then limx→af(x)g(x)=limx→af′(x)g′(x), assuming the limit on the right exists or is ∞ or −∞
- https://math.libretexts.org/Bookshelves/Analysis/Introduction_to_Mathematical_Analysis_I_(Lafferriere_Lafferriere_and_Nguyen)/04%3A_Differentiation/4.04%3A_Section_4-\[\begin{align*} \displaystyle \lim _{x \rightarrow \bar{x}} \frac{f^{\prime}(x)}{g^{\prime}(x)} & =\displaystyle \lim _{x \rightarrow 0} \frac{2+\cos x}{2 x+3} \\[4pt] &=\frac{\displaystyle \lim _{x ...limx→ˉxf′(x)g′(x)=limx→02+cosx2x+3=limx→02+limx→0cosxlimx→02x+3=2+13=1.
- https://math.libretexts.org/Bookshelves/Calculus/Supplemental_Modules_(Calculus)/Integral_Calculus/3%3A_L'Hopital's_Rule_and_Improper_Integrals/3.2%3A_L'Hopital's_Rule\[\begin{align*} \lim_{x\to{\infty}}\big(\arctan x -\dfrac{\pi}{2}e^x\big) &= \lim_{x\to{\infty}} \Big(\dfrac{\arctan x -\dfrac{\pi}{2}}{e^{-x}}\Big) \\[4pt] &=\lim_{x\to{\infty}} \dfrac{\dfrac{1}{1+x...limx→∞(arctanx−π2ex)=limx→∞(arctanx−π2e−x)=limx→∞11+x2−e−x=−limx→∞ex1+x2=−limx→∞ex2x=−limx→∞ex2=−∞.
