Esercizi svolti sui limiti notevoli
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.
Per i richiami teorici più completi si rimanda alla dispensa di teoria sui limiti notevoli. Come ulteriore materiale, si consiglia la lettura degli esercizi sulle forme indeterminate e degli esercizi misti sui limiti.
Scarica gli esercizi svolti
Ottieni il documento contenente 100 esercizi svolti sui limiti notevoli.
Autori e revisori
Mostra autori e revisori.
Testi degli esercizi sui limiti notevoli
. 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.png "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.png "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.png "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.png "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.png "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.png "Rendered by QuickLaTeX.com")
Svolgimento esercizio 6 sui limiti notevoli.
. Calcolare, se esistono, i seguenti limiti applicando solo i limiti notevoli:
![\[\begin{aligned} &1. \quad \lim\limits_{x \to 0}\dfrac{3 \sin x}{4 \ln(1+x)};\\[6pt] &2. \quad \lim\limits_{x \to 0} \dfrac{e^x+e^{-x}-2}{3x^2};\\[6pt] &3. \quad \lim\limits_{x \to \frac{\pi}{2}} \dfrac{\cos x - e^x}{\sin x};\\[6pt] &4. \quad \lim\limits_{x \to 0} \dfrac{\cos x}{3^{2\cos x}-1};\\[6pt] &5. \quad \lim\limits_{x \to +\infty} \left(\dfrac{x+2}{x+1}\right)^x. \end{aligned}\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-079db5b2563101accc2495d741b6cd38_l3.png "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.png "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-5a0865cf58d329aabc180045d62994f0_l3.png "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.png "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.png "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.png "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.png "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.png "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.png "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.png "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.png "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.png "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.png "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.png "Rendered by QuickLaTeX.com")