Benvenuti nella raccolta di esercizi svolti sui limiti notevoli, un concetto fondamentale dell’analisi matematica. Troverete una selezione accurata di 100 esercizi, di vari livelli di difficoltà. Ciascun esercizio è stato scelto per illustrare un’idea o una tecnica utile, che potete trovare nelle soluzioni dettagliate proposte.
Auguriamo una piacevole lettura e un proficuo approfondimenti sia ai neofiti della matematica che agli esperti.
Si vedano i richiami sui limiti notevoli per un ripasso veloce oppure l’articolo di Teoria sui limiti per un riferimento completo di tutte le dimostrazioni.
Come ulteriore materiale, si consiglia la lettura degli esercizi sulle forme indeterminate e degli esercizi misti sui limiti.
Esercizi svolti sui limiti notevoli: autori e revisori
Mostra autori e revisori.
Esercizi svolti sui limiti notevoli: testi
. Calcolare, se esistono, i seguenti limiti applicando solo i limiti notevoli:
![\[\begin{aligned} &1.\quad \lim\limits_{x \to 0} \dfrac{2 \tan x + x}{x};\\[6pt] & 2.\quad \lim\limits_{x \to 0} \dfrac{\sin x + 5x}{x + 2 \sin x};\\[6pt] &3.\quad \lim\limits_{x \to 0} \dfrac{2x \sin x}{\tan^2 x};\\[6pt] &4.\quad \lim\limits_{x \to 0} \dfrac{2 \sin x + 5x}{3 \sin x - x};\\[6pt] &5.\quad \lim\limits_{x \to 0} \dfrac{\sin x - 2x}{x}. \end{aligned}\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-2e4d58dc308224ba83eff5b1fd81787d_l3.svg "Rendered by QuickLaTeX.com")
Svolgimento esercizio 1 sui limiti notevoli.
. Calcolare, se esistono, i seguenti limiti applicando solo i limiti notevoli:
![\[\begin{aligned} &1. \quad \lim\limits_{x \to 0} \dfrac{x^2}{\sin x};\\[6pt] &2. \quad \lim\limits_{x \to 0} \dfrac{\sin 2x + x}{x + \sin x};\\[6pt] &3. \quad \lim\limits_{x \to 0} \dfrac{x^2 + x}{2x+\sin x};\\[6pt] &4. \quad \lim\limits_{x \to \frac{\pi}{2}} \dfrac{\cos x }{1- \sin x };\\[6pt] &5. \quad \lim\limits_{x \to +\infty} \left(\dfrac{x^2-2x+1}{x^2-4x+7}\right)^{\frac{x^2-x+3}{2x^2 + 5}}. \end{aligned}\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-f60b66c9fb27c4a37ab349f965ab995f_l3.svg "Rendered by QuickLaTeX.com")
Svolgimento esercizio 2 sui limiti notevoli.
. Calcolare, se esistono, i seguenti limiti applicando solo i limiti notevoli:
![\[\begin{aligned} &1. \quad \lim\limits_{x \to +\infty} \left( 1 + \dfrac{1}{x^x}\right)^{(x+1)^x};\\[6pt] &2. \quad \lim\limits_{x \to +\infty}\left( \dfrac{x^4+5x^2+5}{(x^2+1)(x^2+2)}\right)^{x^2};\\[6pt] &3. \quad \lim\limits_{x \to 0} \dfrac{\sin^2 x - 8^x + e^{-x}}{x};\\[6pt] &4. \quad \lim\limits_{x \to 0^+} \dfrac{\sqrt{1-\cos x }}{x };\\[6pt] &5. \quad \lim\limits_{x \to 0} \dfrac{1- \cos x - \sin x}{x}. \end{aligned}\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-2ed64742d202cf6ccef4563c6df9244b_l3.svg "Rendered by QuickLaTeX.com")
Svolgimento esercizio 3 sui limiti notevoli.
. Calcolare, se esistono, i seguenti limiti applicando solo i limiti notevoli:
![\[\begin{aligned} &1. \quad \lim_{x \to +\infty} \left(1+\dfrac{3}{2\sqrt{x}}\right)^x;\\[6pt] &2. \quad \lim_{x \to 0} \dfrac{1-\cos x - \sin x}{x^2};\\[6pt] &3. \quad \lim_{x \to 0} \dfrac{\sin^2 2x}{x \; \tan x};\\[6pt] &4. \quad \lim_{x \to \alpha} \dfrac{\sin x - \sin \alpha}{x-\alpha}, \qquad \alpha \in \mathbb{R};\\[6pt] &5. \quad \lim_{x \to \frac{\pi}{6}} \dfrac{2\cos x - \sqrt{3}}{6x-\pi}. \end{aligned}\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-04cfababa840160d10493eb2f50f6d73_l3.svg "Rendered by QuickLaTeX.com")
Svolgimento esercizio 4 sui limiti notevoli.
. Calcolare, se esistono, i seguenti limiti applicando solo i limiti notevoli:
![\[\begin{aligned} &1. \quad \lim\limits_{x \to 3} \dfrac{x-3}{\ln \left(\frac{x}{3}\right)};\\[6pt] &2. \quad \lim\limits_{x \to +\infty} \left(\dfrac{3x-2}{3x+3}\right)^{2x-1};\\[6pt] &3. \quad\lim\limits_{x \to 0^+} \dfrac{x^3}{1-\cos x + \sin \frac{x}{2}};\\[6pt] &4. \quad \lim\limits_{x \to +\infty} \left(\dfrac{5x+3}{5x-1}\right)^{2x-1};\\[6pt] &5. \quad \lim\limits_{x \to \frac{\pi}{6}} \dfrac{2\sin x - 1}{2 \sin(2x)-\sqrt{3}}. \end{aligned}\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-e99b8bfacd575a4350a51c0e44921d03_l3.svg "Rendered by QuickLaTeX.com")
Svolgimento esercizio 5 sui limiti notevoli.
. Calcolare, se esistono, i seguenti limiti applicando solo i limiti notevoli:
![\[\begin{aligned} &1. \quad \lim\limits_{x \to 0} \dfrac{x^3}{2 \sin^2 \frac{x}{2} + \sin \frac{x}{2}};\\[6pt] &2. \quad \lim\limits_{x \to 0} \dfrac{\ln(1+x)}{\sin(2x) + \sin x};\\[6pt] &3. \quad \lim\limits_{x \to 0} \dfrac{2 \sin(1-e^x)}{e^x-1};\\[6pt] &4. \quad \lim\limits_{x \to 0} \dfrac{2^{3x}-1}{2x};\\[6pt] &5. \quad \lim\limits_{x \to 0} \dfrac{\cos(2x)-\cos x}{\cos x-1}. \end{aligned}\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-3b858883c22ecffb41eac85e894b5c40_l3.svg "Rendered by QuickLaTeX.com")
Svolgimento esercizio 6 sui limiti notevoli.
. Calcolare, se esistono, i seguenti limiti applicando solo i limiti notevoli:
![\[\begin{aligned} &1. \quad \lim\limits_{x \to 0}\dfrac{3 \sin x}{4 \ln(1+x)};\\[6pt] &2. \quad \lim\limits_{x \to 0} \dfrac{e^x+e^{-x}-2}{3x^2};\\[6pt] &3. \quad \lim\limits_{x \to 0} \dfrac{\cos x - e^x}{\sin x};\\[6pt] &4. \quad \lim\limits_{x \to \frac{\pi}{2}} \dfrac{\cos x}{3^{2\cos x}-1};\\[6pt] &5. \quad \lim\limits_{x \to +\infty} \left(\dfrac{x+2}{x+1}\right)^x. \end{aligned}\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-b461e45b0a5c5539fbdef8330a20f4b5_l3.svg "Rendered by QuickLaTeX.com")
Svolgimento esercizio 7 sui limiti notevoli.
. Calcolare, se esistono, i seguenti limiti applicando solo i limiti notevoli:
![\[\begin{aligned} &1. \quad \lim\limits_{x \to +\infty} \left(\dfrac{3x-1}{3x+2}\right)^{\frac{x}{2}};\\[6pt] &2. \quad \lim\limits_{x \to \frac{\pi}{2}} (1-\cos x)^{\tan x};\\[6pt] &3. \quad\lim\limits_{x \to +\infty}\left(\dfrac{x+1}{2x-1}\right)^{\frac{x^2-1}{x}};\\[6pt] &4. \quad \lim\limits_{x \to +\infty} \left(\dfrac{3\sqrt{x}-1}{3\sqrt{x}+2}\right)^{\sqrt{x}};\\[6pt] &5. \quad \lim\limits_{x \to +\infty} \left(\dfrac{2x+4}{2x+3}\right)^{x-3}. \end{aligned}\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-438ed883b99de7a2ddcfca101926a157_l3.svg "Rendered by QuickLaTeX.com")
Svolgimento esercizio 8 sui limiti notevoli.
. Calcolare, se esistono, i seguenti limiti applicando solo i limiti notevoli:
![\[\begin{aligned} &1. \quad \lim\limits_{x \to +\infty} \left(1-\dfrac{3}{2x+1}\right)^{x};\\[6pt] &2. \quad \lim\limits_{x \to 0^+} \dfrac{1-\sqrt{\cos x}}{x^2};\\[6pt] &3. \quad \lim\limits_{x \to \frac{\pi}{4}} \dfrac{\cos(2x)}{\cos x - \cos \frac{\pi}{4}};\\[6pt] &4. \quad \lim\limits_{x \to- \alpha} \dfrac{\sin(x+\alpha)}{\cos^2x-\cos^2\alpha}, \qquad \alpha\in \mathbb{R};\\[6pt] &5. \quad \lim\limits_{x \to 0} \dfrac{(1+2x)^4-1}{x}. \end{aligned}\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-96058d22515f91c0eed63e1f3e7bfcfe_l3.svg "Rendered by QuickLaTeX.com")
Svolgimento esercizio 9 sui limiti notevoli.
. Calcolare, se esistono, i seguenti limiti applicando solo i limiti notevoli:
![\[\begin{aligned} &1. \quad \lim\limits_{x \to 1^+} \dfrac{e^{x-1}-1}{1-\cos(1-x)};\\[6pt] &2. \quad \lim\limits_{x \to 0} \dfrac{1-\cos x}{\ln(1+\tan^2x)};\\[6pt] &3. \quad \lim\limits_{x \to 0} \dfrac{3^{\sin x}-1}{x};\\[6pt] &4. \quad \lim\limits_{x \to \pi} \dfrac{\cos x + \cos(2x)}{(x-\pi)^2};\\[6pt] &5. \quad \lim\limits_{x \to +\infty} \left(\dfrac{x}{x+1}\right)^{2x+1}. \end{aligned}\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-d3dd2df7e9918bb2284034c8e0ddd7ce_l3.svg "Rendered by QuickLaTeX.com")
![\[\begin{aligned} &1. \quad \lim\limits_{x \to 0} \dfrac{\sin(3x)(1-\cos x)}{x^2 \sin(6x)};\\[6pt] &2. \quad \lim\limits_{x \to 0^+} \dfrac{\sin(x^2+x)}{x^2};\\[6pt] &3. \quad\lim\limits_{x \to +\infty} \dfrac{5}{x} \; \left(\ln(1+x)-\ln x\right);\\[6pt] &4. \quad \lim\limits_{x \to 0} \dfrac{(1+x^2-x)^{\sqrt{2}}-1}{x};\\[6pt] &5. \quad \lim\limits_{x \to +\infty} \left(\dfrac{3x-4}{3x+2}\right)^{\frac{x+1}{3}} . \end{aligned}\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-f21841615abfb70be8294b0028d235ea_l3.svg "Rendered by QuickLaTeX.com")
![\[\begin{aligned} &1. \quad \lim\limits_{x \to 1} \dfrac{\ln(7x-6)}{\ln(3x-2)};\\[6pt] &2. \quad\lim\limits_{x \to 0} \dfrac{e^{2x}-e^x}{\ln(1+2x)};\\[6pt] &3. \quad \lim\limits_{x \to 4} \dfrac{4^{x-1}-64}{2(x^2-3x-4)};\\[6pt] &4. \quad \lim\limits_{x \to 0^+} \left( \ln x - \ln(\sin(2x)) \right);\\[6pt] &5. \quad \lim\limits_{x \to 0^+} \left(\dfrac{\sin(2x)}{x}\right)^{x+1} . \end{aligned}\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-03bbfde4bf08db3110ecb2cd86274099_l3.svg "Rendered by QuickLaTeX.com")
![\[\begin{aligned} &1. \quad \lim\limits_{x \to 0} \dfrac{1-\cos(3x)}{x\sin x};\\[6pt] &2. \quad \lim\limits_{x \to 0} \dfrac{2x+\sin(3x)}{4x + \sin (7x)};\\[6pt] &3. \quad \lim\limits_{x \to 0} \dfrac{e^x-e^{-x}}{e^{2x}-e^{-2x}};\\[6pt] &4. \quad \lim\limits_{x \to 0} \dfrac{2 \sin x + 5x}{3 \sin x - x};\\[6pt] &5. \quad \lim\limits_{x \to 0} \dfrac{\arcsin(6x)}{\arctan(5x)} . \end{aligned}\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-51c749a17a3dfb8ea8ebc8df41d15162_l3.svg "Rendered by QuickLaTeX.com")
![\[\begin{aligned} &1. \quad \lim\limits_{x \to 0} \dfrac{(1+4x^2)^3-1}{x^2};\\[6pt] &2. \quad \lim\limits_{x \to 0} \dfrac{\sqrt[5]{1+x}-1}{x};\\[6pt] &3. \quad \lim\limits_{x \to 0} \dfrac{\ln(1-x)^2}{x};\\[6pt] &4. \quad \lim\limits_{x \to +\infty} x\left[ \ln(x^2+4)-2\ln x\right] ;\\[6pt] &5. \quad \lim\limits_{x \to 0} \dfrac{(1+2x)^5-1}{5x}. \end{aligned}\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-1e50f89102ebddf6ca38aa617158f27e_l3.svg "Rendered by QuickLaTeX.com")
![\[\begin{aligned} &1. \quad \lim\limits_{x \to +\infty} \dfrac{(1-x)^{2x}}{(1+x^2)^x};\\[6pt] &2. \quad \lim\limits_{x \to e} \dfrac{\ln x^2-2}{x-e};\\[6pt] &3. \quad \lim\limits_{x \to \frac{\pi}{2}} \dfrac{\sin x-1}{\cos x \,\left( \cos\frac{x}{2}-\sin\frac{x}{2}\right) };\\[6pt] &4. \quad \lim\limits_{x \to 0} \dfrac{e^{-x}+\sin x-\cos x}{x};\\[6pt] &5. \quad \lim\limits_{x \to 0} \dfrac{\sqrt[4]{1+x^3}-1}{x^3-x^4}. \end{aligned}\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-a8b9f84434a47c02a57f476dba6147f6_l3.svg "Rendered by QuickLaTeX.com")
. Calcolare, se esistono, i seguenti limiti applicando solo i limiti notevoli:
![\[\begin{aligned} &1. \quad \lim\limits_{x \to 0} \dfrac{e^{\sin(2x)}-e^{\sin x}}{\tan x};\\[6pt] &2. \quad \lim\limits_{x \to +\infty} 2^{-x}\cdot\left( 2+\dfrac{3}{x}\right) ^x;\\[6pt] &3. \quad \lim\limits_{x \to +\infty} x\left[ \ln(2x+1)-\ln x-\ln 2\right] ;\\[6pt] &4. \quad \lim\limits_{x \to 0} \dfrac{e^x-e^{-x}}{\ln(1+x)};\\[6pt] &5. \quad \lim\limits_{x \to 0} \dfrac{\sin^2 x+3e^x-3+x^3}{\ln(1+x)^2+2x-\cos x+1}. \end{aligned}\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-fb45a8c61d3dcfc2cccd7453f82a7dbf_l3.svg "Rendered by QuickLaTeX.com")
![\[\begin{aligned} &1. \quad \lim\limits_{x \to 0} \dfrac{1-\sqrt{\cos x}}{\sqrt{1-\cos x}};\\[6pt] &2. \quad \lim\limits_{x \to 0^+} \dfrac{ \left( \sin x\right) ^{ \sin x}-1}{ \sqrt{\ln\left( \dfrac{1}{\cos x}\right) }};\\[6pt] &3. \quad \lim\limits_{x \to 0^+} \dfrac{ \left( \cos x\right) ^{ \cos x}-1}{ \left( \sin x\right) ^{ \sin x}-1};\\[6pt] &4. \quad \lim\limits_{x \to 0^+} \dfrac{\left( \sin 2x\right) ^{ \sin x}-1}{\left( \sin x\right) ^{ \sin 2x}-1};\\[6pt] &5. \quad \lim\limits_{x \to \frac{\pi}{2}} \dfrac{\cos x}{1-\sin x}. \end{aligned}\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-e0bd8132e2e237de7452a67802cccba0_l3.svg "Rendered by QuickLaTeX.com")
. Calcolare, se esistono, i seguenti limiti applicando solo i limiti notevoli:
![\[\begin{aligned} &1. \quad \lim\limits_{x \to 0} \dfrac{\left( e^{\sin x}-1\right)\cdot \tan x}{1-\cos x};\\[6pt] &2. \quad \lim\limits_{x \to 0} \dfrac{\left( e^{5x^2}-1\right) \cdot\ln^2\left( 1+3x\right) }{1-\cos\left( x^2\right) };\\[6pt] &3. \quad \lim\limits_{x \to 0} \dfrac{\left( 4^x-1\right) \cdot\log_2\left( \cos x\right) }{\sqrt[9]{1+9x^3}-1};\\[6pt] &4. \quad \lim\limits_{x \to 0} \dfrac{\left( \sqrt[9]{\cos(6x)}-1\right)\cdot \arctan x}{\left( e^{\cos x}-e\right) \cdot\ln(1+\sin x)};\\[6pt] &5. \quad \lim\limits_{x \to 0} \dfrac{(1+2x)^{5x^2}-1}{(1+3x)^{4x^2}-1}. \end{aligned}\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-9ad248a9a86f5918b6304616e63a0b4d_l3.svg "Rendered by QuickLaTeX.com")
![\[\begin{aligned} &1. \quad \lim\limits_{x \to 0^+} \dfrac{\pi-2\arctan\dfrac{1}{x^3}}{\tan(2x)-\sin(2x)};\\[6pt] &2. \quad \lim\limits_{x \to 0} \dfrac{\ln(\cos x)+\ln(2-\cos x) }{x\cdot(\tan x-\sin x) };\\[6pt] &3. \quad \lim\limits_{x \to 0^+} \dfrac{\left( \cos\sqrt{x}\right) ^{\sin x}-1 }{\sqrt[3]{\cos x}-1};\\[6pt] &4. \quad \lim\limits_{x \to \pi} \dfrac{\sin^2x}{\ln(2+\cos x)};\\[6pt] &5. \quad \lim\limits_{x \to +\infty} \dfrac{\sin\left( 1-\cos\dfrac{1}{x}\right) \cdot\arctan(1-x)}{e^{\frac{1}{x^2}}-1}. \end{aligned}\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-9ea0d5194f9a645baf9e406dd0a1bcd1_l3.svg "Rendered by QuickLaTeX.com")
. Calcolare, se esistono, i seguenti limiti applicando solo i limiti notevoli:
![\[\begin{aligned} &1. \quad \lim\limits_{x \to +\infty} \left(\dfrac{4^{\frac{1}{x^2}}+6^{\frac{1}{x^2} } }{3^{\frac{1}{x^2}}+5^{\frac{1}{x^2}}}\right)^{x^2};\\[6pt] &2. \quad \lim\limits_{x \to 0} \dfrac{\ln(\cos(\alpha x))}{\ln(\cos(\beta x))}, \qquad \alpha \in \mathbb{R}, \beta \in \mathbb{R}\setminus\{0\} ;\\[6pt] &3. \quad \lim\limits_{x \to 0^+} (\text{settsinh}(x))^{\cot x};\\[6pt] &4. \quad \lim\limits_{x \to 0^+} \dfrac{(1+\alpha x)^{\frac{1}{x}} + \sqrt{ \sin(\pi x^\alpha)}}{ 1-\cos(3 \sqrt{x})}, \qquad \alpha >0 ;\\[6pt] &5. \quad \lim\limits_{x \to +\infty} \dfrac{\ln\left(1+\frac{1}{x^3}\right)^4}{\left(\text{arcsin}\left(\frac{1}{x}\right)\right)^3 - \frac{1}{x^\alpha}\sin\left( \frac{1}{x}\right)}, \qquad \alpha > -1. \end{aligned}\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-48ebf065b3529c8ba1ead849f49f8833_l3.svg "Rendered by QuickLaTeX.com")