{"id":280913,"date":"2025-08-05T08:18:41","date_gmt":"2025-08-05T08:18:41","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=280913"},"modified":"2025-08-05T08:18:43","modified_gmt":"2025-08-05T08:18:43","slug":"plk-fxx2-2-gx3x2-tentokon","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/08\/05\/plk-fxx2-2-gx3x2-tentokon\/","title":{"rendered":"PlK F(x)=x^2-2 g(x)=3x+2 tentokon"},"content":{"rendered":"\n
\"\"<\/figure>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Based on the handwritten notes in the image, here is the transcription of the mathematical problems and their solutions.<\/p>\n\n\n\n

Problem 1<\/strong>
lim (x\u2192\u221e) [\u221a(x\u00b2 + x + 1) – \u221a(x\u00b2 – x + 1)]
= 1<\/p>\n\n\n\n

Cara Cepat (Quick Method) for Problem 1<\/strong>
Given the form lim (x\u2192\u221e) [\u221a(ax\u00b2 + bx + c) – \u221a(px\u00b2 + qx + r)], if a = p, the limit is (b – q) \/ (2\u221aa).
Here, a=1, b=1, c=1 and p=1, q=-1, r=1.
Since a=p, the formula applies:
(1 – (-1)) \/ (2\u221a1) = 2 \/ 2 = 1<\/p>\n\n\n\n

Problem 2<\/strong>
lim (x\u21920) [x tan(3x) \/ sin\u00b2(6x)]
= 1\/12<\/p>\n\n\n\n

Problem 3<\/strong>
lim (x\u2192\u221e) [(2x\u00b2 + 3x) \/ \u221a(x\u2074 – x)]
= 2<\/p>\n\n\n\n

Explanation<\/h3>\n\n\n\n

The provided image displays handwritten solutions to three distinct calculus problems involving limits. Each solution employs standard techniques appropriate for the form of the limit.<\/p>\n\n\n\n

The first problem is a limit at infinity that presents an indeterminate form of \u221e minus \u221e. The standard approach, shown in the notes, is to multiply the expression by its conjugate, which is \u221a(x\u00b2 + x + 1) + \u221a(x\u00b2 – x + 1). This algebraic manipulation removes the square roots from the numerator, leaving a simplified expression of 2x. The next step involves dividing both the numerator and denominator by the highest power of x, which is x. As x approaches infinity, terms like 1\/x and 1\/x\u00b2 approach zero. This simplification leads to the final result of 2 \/ (\u221a1 + \u221a1), which equals 1. The notes also show a shortcut formula for this specific type of limit, confirming the same answer.<\/p>\n\n\n\n

The second problem is a limit as x approaches zero involving trigonometric functions, resulting in the indeterminate form 0\/0. The strategy used is to rewrite the expression sin\u00b2(6x) as sin(6x) * sin(6x) and then separate the fraction into a product of two simpler limits: [x \/ sin(6x)] and [tan(3x) \/ sin(6x)]. The solution then applies fundamental trigonometric limit identities, where lim (x\u21920) x\/sin(ax) = 1\/a and lim (x\u21920) tan(ax)\/sin(bx) = a\/b. Applying these rules, the limits of the individual parts are 1\/6 and 3\/6, respectively. Multiplying these results gives the final answer of 1\/12.<\/p>\n\n\n\n

The third problem is another limit at infinity, this time with a rational expression. To solve this, one must identify the highest power of x in the overall expression. In the denominator, \u221a(x\u2074) is equivalent to x\u00b2. Therefore, both the numerator and the denominator are divided by x\u00b2. This transforms the expression into (2 + 3\/x) \/ \u221a(1 – 1\/x\u00b3) As x approaches infinity, the terms 3\/x and 1\/x\u00b3 go to zero, simplifying the expression to 2 \/ \u221a1, which results in the final answer of 2.<\/p>\n\n\n\n

\"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

The Correct Answer and Explanation is: Based on the handwritten notes in the image, here is the transcription of the mathematical problems and their solutions. Problem 1lim (x\u2192\u221e) [\u221a(x\u00b2 + x + 1) – \u221a(x\u00b2 – x + 1)]= 1 Cara Cepat (Quick Method) for Problem 1Given the form lim (x\u2192\u221e) [\u221a(ax\u00b2 + bx + c) […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-280913","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/280913","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=280913"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/280913\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=280913"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=280913"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=280913"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}