{"id":21228967,"url":"https://github.com/quantum-software-development/limits-calculus-i","last_synced_at":"2025-07-28T21:08:55.017Z","repository":{"id":241446036,"uuid":"805098985","full_name":"Quantum-Software-Development/Limits-Calculus-I","owner":"Quantum-Software-Development","description":"✍️ Math Calculus I -  Limits and Derivatives Exercises - AI Data Science - PUCSP University","archived":false,"fork":false,"pushed_at":"2025-07-01T19:26:03.000Z","size":32943,"stargazers_count":2,"open_issues_count":832,"forks_count":0,"subscribers_count":1,"default_branch":"main","last_synced_at":"2025-07-01T20:27:57.961Z","etag":null,"topics":["calculus","derivative","django","latex-code","leonardo-ai","limits","mathpix","maths","numpy","overleaf","python3","rest-api"],"latest_commit_sha":null,"homepage":"https://github.com/Quantum-Software-Development/Math","language":"Shell","has_issues":true,"has_wiki":null,"has_pages":null,"mirror_url":null,"source_name":null,"license":"mit","status":null,"scm":"git","pull_requests_enabled":true,"icon_url":"https://github.com/Quantum-Software-Development.png","metadata":{"files":{"readme":"README.md","changelog":null,"contributing":null,"funding":".github/FUNDING.yml","license":"LICENSE","code_of_conduct":null,"threat_model":null,"audit":null,"citation":null,"codeowners":null,"security":null,"support":"support  Material/Plano de Ensino Matemática.pdf","governance":null,"roadmap":null,"authors":null,"dei":null,"publiccode":null,"codemeta":null,"zenodo":null},"funding":{"github":"Quantum-Software-Development","Custom":"https://github.com/sponsors/Quantum-Software-Development/card"}},"created_at":"2024-05-23T22:06:49.000Z","updated_at":"2025-07-01T19:26:05.000Z","dependencies_parsed_at":"2024-08-29T03:24:38.883Z","dependency_job_id":"51e756b6-5f19-41f8-baff-772f9144b0dc","html_url":"https://github.com/Quantum-Software-Development/Limits-Calculus-I","commit_stats":null,"previous_names":["quantum-software-development/math"],"tags_count":0,"template":false,"template_full_name":null,"purl":"pkg:github/Quantum-Software-Development/Limits-Calculus-I","repository_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/Quantum-Software-Development%2FLimits-Calculus-I","tags_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/Quantum-Software-Development%2FLimits-Calculus-I/tags","releases_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/Quantum-Software-Development%2FLimits-Calculus-I/releases","manifests_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/Quantum-Software-Development%2FLimits-Calculus-I/manifests","owner_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners/Quantum-Software-Development","download_url":"https://codeload.github.com/Quantum-Software-Development/Limits-Calculus-I/tar.gz/refs/heads/main","sbom_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/Quantum-Software-Development%2FLimits-Calculus-I/sbom","host":{"name":"GitHub","url":"https://github.com","kind":"github","repositories_count":267585787,"owners_count":24111577,"icon_url":"https://github.com/github.png","version":null,"created_at":"2022-05-30T11:31:42.601Z","updated_at":"2022-07-04T15:15:14.044Z","status":"online","status_checked_at":"2025-07-28T02:00:09.689Z","response_time":68,"last_error":null,"robots_txt_status":"success","robots_txt_updated_at":"2025-07-24T06:49:26.215Z","robots_txt_url":"https://github.com/robots.txt","online":true,"can_crawl_api":true,"host_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub","repositories_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories","repository_names_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repository_names","owners_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners"}},"keywords":["calculus","derivative","django","latex-code","leonardo-ai","limits","mathpix","maths","numpy","overleaf","python3","rest-api"],"created_at":"2024-11-20T23:23:42.904Z","updated_at":"2025-07-28T21:08:54.992Z","avatar_url":"https://github.com/Quantum-Software-Development.png","language":"Shell","readme":"\n\u003c!--\n \u003cp align=\"center\"\u003e\n\u003cimg src=\"https://github.com/Quantum-Software-Development/Math/assets/113218619/58c8c407-2971-4a65-9030-e25d76617687\"/\u003e\n--\u003e\n\n ## \u003cp align=\"center\"\u003e ✍️ Limits - Calculus I - Resolution of Mathematics Exercises\n\n\u003cbr\u003e\u003cbr\u003e\n\n \u003cp align=\"center\"\u003e\n\u003cimg src=\"https://github.com/Quantum-Software-Development/Math/assets/113218619/ed52f2ce-48f0-41d8-a973-611db792559a\"/\u003e\n\n\u003cbr\u003e\u003cbr\u003e\n\n#\n\n#### \u003cp align=\"center\"\u003e AI Data Science - 1st Semester / 20024 - PUCSP University - Math Repository - [Professor Eric Bacconi Gonçalves](https://www.linkedin.com/in/eric-bacconi-423137/)\n\n#\n\n\u003cbr\u003e\n\n### \u003cp align=\"center\"\u003e [![Sponsor Quantum Software Development](https://img.shields.io/badge/Sponsor-Quantum%20Software%20Development-brightgreen?logo=GitHub)](https://github.com/sponsors/Quantum-Software-Development)\n\n\u003cbr\u003e\n\n## [1. Find the limits:]()\n\n### 1a)  **Limit Expression:**\n\n$$lim_{{x \\to 3}} \\frac{{x^2 - 9}}{{x - 3}}$$\n\n\u003cbr\u003e\n\n**Simplified Form:** The numerator  $$\\large x^2 - 9$$, can be factored as (x + 3)(x - 3) ), which simplifies the expression to:\n\n\u003cbr\u003e\n\n$$\\\\begin{align*}\n\\large \\lim_{{x \\to 3}} (x + 3)\u2028\n\\end{align*}\n\\$$\n\n\u003cbr\u003e\n\n[**Final Result:**]()\n\nSubstituting ( x ) with 3, we get: \n\n$$\\large  3 + 3 = 6$$\n\n\u003cbr\u003e\n\n**Explanation:** The limit as ( x ) approaches 3 for the function $\\large \\frac{{x^2 - 9}}{{x - 3}}$ is 6.\n\nThis is because the factor ( x - 3 ) in the denominator cancels out with the same factor in the numerator, leaving ( x + 3 ) which evaluates to 6 when ( x ) is 3.\n \n#\n\n### 1b) **The Limit Expression given is:**\n\n$$\\lim_{{x \\to -7}} \\frac{{49 - x^2}}{{7 + x}}$$\n\n\u003cbr\u003e\n\n**Simplified Form:** The numerator $\\large ( 49 - x^2 )$ is a difference of squares and can be factored as $\\large (7 + x)(7 - x)$.\n\n\u003cbr\u003e\n\n**This allows us to simplify the expression by canceling out the common factor of:**  $\\large ( 7 + x )$ in the numerator and denominator:\n\n$$\\large \\lim_{{x \\to -7}} (7 - x)$$\n\n\u003cbr\u003e\n\n[**Final Result:**]()\n\nWhen we substitute ( x ) with -7, the expression simplifies to:\u20287 - (-7) = 14\n\n \u003cbr\u003e\n\n**Explanation:** The limit of the function $(\\Large \\frac{{49 - x^2}}{{7 + x}} )$ as ( x ) approaches -7 is 14.\n\nThis result is obtained because after canceling the common factor, we are left with ( 7 - x ), which equals 14 when ( x ) is -7.\n\n  #\n\n### 1c) **Limit Expression:**\n\n$$\n\\lim_{{x \\to 1}} \\frac{{x^2 - 4x + 3}}{{x - 1}}\n$$\n\n[Solution]()\n\nTo solve the limit, we can factor the numerator:\n\n$$\nx^2 - 4x + 3 = (x - 1)(x - 3)\n$$\n\nSo the limit becomes:\n\n$$\n\\lim_{{x \\to 1}} \\frac{{(x - 1)(x - 3)}}{{x - 1}}\n$$\n\nWe can cancel out the \\((x - 1)\\) terms:\n\n$$\n\\lim_{{x \\to 1}} (x - 3)\n$$\n\nNow, we can directly substitute \\( x = 1 \\):\n\n$$\n1 - 3 = -2\n$$\n\nTherefore, the limit is:\n\n$$\n\\lim_{{x \\to 1}} \\frac{{x^2 - 4x + 3}}{{x - 1}} = -2\n$$\n\n#\n\n### 1d)   **Limit Expression:**\u2028\n\n$$\\lim_{{x \\to 1}} \\frac{{x^2 - 2x + 1}}{{x - 1}}$$\n\n\u003cbr\u003e\n\nTo calculate the limit, we can simplify the expression by factoring the numerator, which is a perfect square trinomial. Factoring (x^2 - 2x + 1), we get ((x - 1)(x - 1)). The denominator is already in factored form as (x - 1). Thus, the function simplifies to:\n\n$$\\frac{(x - 1)(x - 1)}{x - 1}$$\n\nAfter canceling out the common term ( x - 1 ), we are left with:\n\n$$\\lim_{{x \\to 1}} (x - 1)$$\n\nSince there are no more terms that depend on ( x ), this simplifies to:\n\n$$\\lim_{{x \\to 1}} = x - 1 = 0$$\n\n[Final Result:]()\n\nThe limit of the function as ( x ) approaches 1 is simply 0.\n\n#\n\n### 1e)  **Limit Expression:**\u2028\n\n$$\\lim_{{x \\to 2}} \\frac{{x - 2}}{{x^2 - 4}}$$\n\n\u003cbr\u003e\n\nTo solve this limit, we need to factor the denominator and simplify the expression. The denominator ( x^2 - 4 ) can be factored into ( (x + 2)(x - 2) ), which allows us to cancel out the ( x - 2 ) term in the numerator:\n\n$$\\lim_{{x \\to 2}} \\frac{1}{{x + 2}}$$\n\nSubstituting ( x = 2 ) into the simplified expression, the final value:\n\n$$\\frac{1}{4}$$\n\n\u003cbr\u003e\n\n[Final Result:]()\n\nThe limit of the function as ( x ) approaches 1 is simply $$\\frac{1}{4}$$\n\n#\n\n### 1f) **Limit Expression:**\u2028\n\n$$\\(\\lim_{{x \\to 3}} \\frac{{x^3 - 27}}{{x^2 - 5x + 6}}\\)$$\n\n\u003cbr\u003e\n\nThis limit can be solved using factorization and polynomial division: \u003cbr\u003e\u003cbr\u003e\n\n$$\\\n\\begin{align*}\n\\lim_{{x \\to 3}} \\frac{{x^3 - 27}}{{x^2 - 5x + 6}} \n\u0026= \\lim_{{x \\to 3}} \\frac{{(x - 3)(x^2 + 3x + 9)}}{{(x - 2)(x - 3)}} \\\\\n\u0026= \\lim_{{x \\to 3}} \\frac{{x - 3}}{{x - 2}} \\\\\n\u0026= \\frac{{3 - 3}}{{3 - 2}} \\\\\n\u0026= 1\n\\end{align*}\n\\$$\n\n\u003cbr\u003e\n\n[Final Result:]()\n\nThe limit of the function as ( x ) approaches 1 is simply $$\\frac{1}{4}$$\n\n#\n\n### 1g:  **Limit Expression:**\u2028\n\n$$\\(\\lim_{{x \\to \\infty}} \\frac{{x^2}}{{2x^2 - x}}\\)$$\n\n\u003cbr\u003e\n\nIn this case, we can use L'Hôpital's rule, as the limit is of the form \\(\\frac{0}{0}\\) or \\(\\frac{\\infty}{\\infty}\\) when \\(x\\) tends to infinity.\n\n$$\\\n\\begin{align*}\n\\lim_{{x \\to \\infty}} \\frac{{x^2}}{{2x^2 - x}} \u0026= \\lim_{{x \\to \\infty}} \\frac{{\\frac{d}{dx}[x^2]}}{{\\frac{d}{dx}[2x^2 - x]}} \\\\\n\u0026= \\lim_{{x \\to \\infty}} \\frac{{2x}}{{4x - 1}} \\\\\n\u0026= \\lim_{{x \\to \\infty}} \\frac{{2}}{{4 - \\frac{1}{x}}} \\\\\n\u0026= \\frac{2}{4} \\\\\n\u0026= \\frac{1}{2}\n\\end{align*}\n\\$$\n\n[Final Result:]() \n\nThe limit of the expression is $$\\frac{1}{2}$$\n\n\u003cbr\u003e\n\n## [2. Solve the Limits:]()\n\n### 2a):  **Limit Expression:**\u2028\n\n $\\lim_{x \\to \\infty} \\frac{1}{x^2}$ \n\n\u003cbr\u003e \n\nThe limit as ( x ) approaches infinity for ( $\\frac{1}{{x^2}}$ ):\n\nis:\u2028$\\lim_{{x \\to \\infty}} \\frac{1}{{x^2}} = 0$\u2028\n\nAs ( x ) increases without bound, the value of ( \\frac{1}{{x^2}} ) approaches 0 because the denominator grows much faster than the numerator.\n\n#\n\n### 2b)  **Limit Expression:**\u2028\n\n( $\\lim_{x \\to -\\infty} \\frac{1}{x^2}$ )\n\n\u003cbr\u003e\n\nThe limit as ( x ) approaches negative infinity for ( $\\frac{1}{{x^2}}$ ) is:\u2028\n\n $\\lim_{x \\to -\\infty} \\frac{1}{x^2}$ = 0\n\nAs ( x ) decreases without bound, the value of ( $\\frac{1}{{x^2}}$ ) approaches 0, similar to part a), because squaring a negative number results in a positive number, which grows larger.\n\n#\n\n### 2c)  **Limit Expression:**\u2028\n\n$\\lim_{x \\to \\infty} x^4$\u2028\n\n\u003cbr\u003e\n\nThe limit as ( x ) approaches infinity for ( x^4 ) is:\u2028grows at an increasing rate and approaches infinity for ( x^4 ) is:\n\n$\\lim_{x \\to \\infty} x^4$\u2028 = $\\infty$\n\nSimilar to the previous expressions, the term ( 2x^5 ) grows at a faster rate than the others, causing the expression to approach infinity.\n\n#\n\n### 2d)  **Limit Expression:**\u2028\n\n$\\lim_{{x \\to \\infty}}$ $(2x^4 - 3x^3 + x + 6)$ \n\n\u003cbr\u003e\n\nThe limit as ( x ) approaches negative infinity for ( 2x^4 - 3x^3 + x + 6 ) is:\n\n$\\lim_{x \\to -\\infty} (2x^4 - 3x^3 + x + 6) = \\infty$\u2028\n\nEven though ( x ) is negative, the highest power term ( x^4 )  will still lead the expression to increase without bound because the even power makes it positive.\n\n#\n\n### 2e)  **Limit Expression:**\u2028\n\n2x^5 - 3x^2 + 6\n\n\u003cbr\u003e\n\nThe limit as ( x ) approaches infinity for $\\( 2x^5 - 3x^2 + 6 ) is:\u2028\n\nThe limit as ( x ) approaches negative infinity for $\\( 2x^4 - 3x^3 + x + 6 )$ is:\u2028\n\n$\\lim_{x \\to -\\infty} (2x^4 - 3x^3 + x + 6) = \\infty$\n\nEven though ( x ) is negative, y.\n\n\u003cbr\u003e\n\n## [3.Calculate the Following Limits]()\n\n### 3a: Finding the limit of a polynomial function as x approaches infinity\n\nThe given function is a polynomial function of the form: \n\n$$f(x)=axn+bxn−1+cxn−2+...+dx+e$$\n\n\u003cbr\u003e\n\nAs x approaches infinity, the highest power of x in the function dominates the value of the function. This means that we can ignore all the lower-order terms, and simply consider the behavior of the highest-order term.\nIn this case, the highest-order term is 2x4. As x approaches infinity, x4 also approaches infinity, and so the function f(x) also approaches infinity.\n\nTherefore, the limit of the function as x approaches infinity is infinity. We can write this mathematically as:\n\n$$x→∞lim x32x4−3x3+x+6 =0$$\n\n#\n\n### 3b:Finding the limit of a rational function as x approaches infinity\n\nThe given function is a rational function of the form \n\n$$f(x)=cxm+fxm−1+...+gx+haxn+bxn−1+...+dx+e$$\n\n\u003cbr\u003e\n\n, where n \u003e m. As x approaches infinity, the highest power of x in the numerator dominates the value of the numerator, and the highest power of x in the denominator dominates the value of the denominator.  This means that we can ignore all the lower-order terms, and simply consider the behavior of the highest-order terms.\n\nIn this case, the highest-order term in the numerator is 2x4, and the highest-order term in the denominator is x3. \n\nAs x approaches infinity, 2x4 grows much faster than x3, and so the function f(x) approaches zero.\n\n#\n\n### These processes above  demonstrates how limits help us understand the behavior of functions near points that might not be defined, by finding equivalent expressions that are easier to evaluate.\n\n#\n\n###### \u003cp align=\"center\"\u003e Copyright 2024 Quantum Software Development. Code released under the [MIT License license.](https://github.com/Quantum-Software-Development/Math/blob/3bf8270ca09d3848f2bf22f9ac89368e52a2fb66/LICENSE)\n\n\n  \n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","funding_links":["https://github.com/sponsors/Quantum-Software-Development","https://github.com/sponsors/Quantum-Software-Development/card"],"categories":[],"sub_categories":[],"project_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fquantum-software-development%2Flimits-calculus-i","html_url":"https://awesome.ecosyste.ms/projects/github.com%2Fquantum-software-development%2Flimits-calculus-i","lists_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fquantum-software-development%2Flimits-calculus-i/lists"}