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- https://math.libretexts.org/Courses/Coastline_College/Math_C180%3A_Calculus_I_(Everett)/02%3A_Limits/2.05%3A_The_Limit_LawsIn this section, we establish laws for calculating limits and learn how to apply these laws. In the Student Project at the end of this section, you have the opportunity to apply these limit laws to de...In this section, we establish laws for calculating limits and learn how to apply these laws. In the Student Project at the end of this section, you have the opportunity to apply these limit laws to derive the formula for the area of a circle by adapting a method devised by the Greek mathematician Archimedes. We begin by restating two useful limit results from the previous section. These two results, together with the limit laws, serve as a foundation for calculating many limits.
- https://math.libretexts.org/Bookshelves/Calculus/Calculus_3e_(Apex)/01%3A_Limits/1.03%3A_Finding_Limits_AnalyticallyRecognizing that ϵ-δ proofs are cumbersome, this section gives a series of theorems which allow us to find limits much more quickly and intuitively. One of the main results of this section states th...Recognizing that ϵ-δ proofs are cumbersome, this section gives a series of theorems which allow us to find limits much more quickly and intuitively. One of the main results of this section states that many functions that we use regularly behave in a very nice, predictable way. In the next section we give a name to this nice behavior; we label such functions as continuous. Defining that term will require us to look again at what a limit is and what causes limits to not exist.
- https://math.libretexts.org/Courses/Reedley_College/Calculus_I_(Casteel)/02%3A_Limits/2.03%3A_The_Limit_LawsIn this section, we establish laws for calculating limits and learn how to apply these laws. In the Student Project at the end of this section, you have the opportunity to apply these limit laws to de...In this section, we establish laws for calculating limits and learn how to apply these laws. In the Student Project at the end of this section, you have the opportunity to apply these limit laws to derive the formula for the area of a circle by adapting a method devised by the Greek mathematician Archimedes. We begin by restating two useful limit results from the previous section. These two results, together with the limit laws, serve as a foundation for calculating many limits.
- https://math.libretexts.org/Courses/Mission_College/Math_3A%3A_Calculus_I_(Reed)/02%3A_Limits/2.03%3A_The_Limit_LawsIn this section, we establish laws for calculating limits and learn how to apply these laws. In the Student Project at the end of this section, you have the opportunity to apply these limit laws to de...In this section, we establish laws for calculating limits and learn how to apply these laws. In the Student Project at the end of this section, you have the opportunity to apply these limit laws to derive the formula for the area of a circle by adapting a method devised by the Greek mathematician Archimedes. We begin by restating two useful limit results from the previous section. These two results, together with the limit laws, serve as a foundation for calculating many limits.
- https://math.libretexts.org/Courses/Borough_of_Manhattan_Community_College/MAT301_Calculus_I/06%3A_professor_playground/6.01%3A_0.0_Special_SymbolsSome symbols
- https://math.libretexts.org/Courses/Monroe_Community_College/MTH_210_Calculus_I_(Professor_Dean)/professor_playground/0.0_Special_SymbolsSome symbols
- https://math.libretexts.org/Courses/City_College_of_San_Francisco/CCSF_Calculus/02%3A_Learning_Limits/2.03%3A_The_Limit_Laws_-_Limits_at_Finite_NumbersThis section introduces the Limit Laws for calculating limits at finite numbers. It covers fundamental rules, including the Sum, Difference, Product, Quotient, and Power Laws, which simplify finding l...This section introduces the Limit Laws for calculating limits at finite numbers. It covers fundamental rules, including the Sum, Difference, Product, Quotient, and Power Laws, which simplify finding limits of functions. The section emphasizes understanding and applying these laws systematically, providing examples and exercises to reinforce learning. These rules are essential for solving more complex limits and serve as a foundation for further study in Calculus.
- https://math.libretexts.org/Bookshelves/Calculus/Calculus_(Guichard)/04%3A_Transcendental_Functions/4.03%3A_A_Hard_LimitBefore we can complete the calculation of the derivative of the sine, we need one other limit: \(\lim_{x\to0}{\cos x - 1\over x}.\) This limit is just as hard as \(\sin x/x\), but closely related to i...Before we can complete the calculation of the derivative of the sine, we need one other limit: \(\lim_{x\to0}{\cos x - 1\over x}.\) This limit is just as hard as \(\sin x/x\), but closely related to it, so that we don't have to a similar calculation; instead we can do a bit of tricky algebra: \[{\cos x - 1\over x}={\cos x - 1\over x}{\cos x+1\over\cos x+1} ={\cos^2 x - 1\over x(\cos x+1)}={-\sin^2 x\over x(\cos x+1)}= -{\sin x\over x}{\sin x\over \cos x + 1}.\]
- https://math.libretexts.org/Bookshelves/Analysis/Introduction_to_Mathematical_Analysis_I_(Lafferriere_Lafferriere_and_Nguyen)/02%3A_Sequences/2.01%3A_Convergence\left|a_{n}-\frac{3}{2}\right| &=\left|\frac{3 n^{2}+4}{2 n^{2}+n+5}-\frac{3}{2}\right|=\left|\frac{2\left(3 n^{2}+4\right)-3\left(2 n^{2}+n+5\right)}{2\left(2 n^{2}+n+5\right)}\right|=\left|\frac{-7-...\left|a_{n}-\frac{3}{2}\right| &=\left|\frac{3 n^{2}+4}{2 n^{2}+n+5}-\frac{3}{2}\right|=\left|\frac{2\left(3 n^{2}+4\right)-3\left(2 n^{2}+n+5\right)}{2\left(2 n^{2}+n+5\right)}\right|=\left|\frac{-7-3 n}{2\left(2 n^{2}+n+5\right)}\right| \\ Since \(\left|a_{n}\right|-|a| \leq \| a_{n}|-| a|| \leq\left|a_{n}-a\right|\), this implies \(\left|a_{n}\right|<1+|a|\) for all \(n \geq N\).
- https://math.libretexts.org/Courses/Mount_Royal_University/Calculus_for_Scientists_I/2%3A_Limit__and_Continuity_of_Functions/1.7%3A_Limit_of__Trigonometric_functionsLet \(P=(x,y)\) be a point on the unit circle and let θ be the corresponding angle . Since the angle \(θ\) and \(θ+2π\) correspond to the same point \(P\), the values of the trigonometric functions at...Let \(P=(x,y)\) be a point on the unit circle and let θ be the corresponding angle . Since the angle \(θ\) and \(θ+2π\) correspond to the same point \(P\), the values of the trigonometric functions at \(θ\) and at \(θ+2π\) are the same. The area of the large triangle is \(\frac12\tan\theta\); the area of the sector is \(\theta/2\); the area of the triangle contained inside the sector is \(\frac12\sin\theta\).
- https://math.libretexts.org/Courses/Lake_Tahoe_Community_College/Interactive_Calculus_Q1/02%3A_Limits/2.04%3A_The_Limit_LawsIn this section, we establish laws for calculating limits and learn how to apply these laws. In the Student Project at the end of this section, you have the opportunity to apply these limit laws to de...In this section, we establish laws for calculating limits and learn how to apply these laws. In the Student Project at the end of this section, you have the opportunity to apply these limit laws to derive the formula for the area of a circle by adapting a method devised by the Greek mathematician Archimedes. We begin by restating two useful limit results from the previous section. These two results, together with the limit laws, serve as a foundation for calculating many limits.
