To estimate the limit lim\u2061h\u21920ln\u2061(1+h)h,\\lim_{h \\to 0} \\frac{\\ln(1 + h)}{h},<\/p>\n\n\n\n
we can use graphs or tables<\/strong>. But first, let\u2019s understand what this limit represents and how we can estimate it.<\/p>\n\n\n\n 1.00\\boxed{1.00}<\/p>\n\n\n\n This limit is a well-known fundamental limit in calculus: lim\u2061h\u21920ln\u2061(1+h)h=1.\\lim_{h \\to 0} \\frac{\\ln(1 + h)}{h} = 1.<\/p>\n\n\n\n We can understand and estimate this using a table of values<\/strong> or a graph<\/strong> of the function f(h)=ln\u2061(1+h)h.f(h) = \\frac{\\ln(1 + h)}{h}.<\/p>\n\n\n\n Let\u2019s look at some values of hh close to 0:<\/p>\n\n\n\n As hh gets closer to 0 from both sides (positive and negative), the values of the function approach 1<\/strong>. This strongly suggests that: lim\u2061h\u21920ln\u2061(1+h)h=1.\\lim_{h \\to 0} \\frac{\\ln(1 + h)}{h} = 1.<\/p>\n\n\n\n This limit is the derivative of ln\u2061(x)\\ln(x) at x=1x = 1. In calculus, the derivative is defined as: f\u2032(a)=lim\u2061h\u21920f(a+h)\u2212f(a)h.f'(a) = \\lim_{h \\to 0} \\frac{f(a + h) – f(a)}{h}.<\/p>\n\n\n\n For f(x)=ln\u2061(x)f(x) = \\ln(x), we get: ln\u2061\u2032(1)=lim\u2061h\u21920ln\u2061(1+h)\u2212ln\u2061(1)h=lim\u2061h\u21920ln\u2061(1+h)h.\\ln'(1) = \\lim_{h \\to 0} \\frac{\\ln(1 + h) – \\ln(1)}{h} = \\lim_{h \\to 0} \\frac{\\ln(1 + h)}{h}.<\/p>\n\n\n\n So, the value of this limit tells us the rate of change<\/strong> of the natural logarithm at 1, and it is exactly 1<\/strong>. This is a key concept in understanding how logarithmic functions behave near 1.<\/p>\n\n\n\n 1.00\\boxed{1.00}<\/p>\n\n\n\n Estimate the following limit using graphs or tables. In (1 + h) lim h h?0 In (1 + h) lim h h- (Type an integer or decimal rounded to the nearest hundredth as needed.) The Correct Answer and Explanation is: To estimate the limit lim\u2061h\u21920ln\u2061(1+h)h,\\lim_{h \\to 0} \\frac{\\ln(1 + h)}{h}, we can use graphs or […]<\/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-219797","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/219797","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=219797"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/219797\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=219797"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=219797"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=219797"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n\n\n\nCorrect Answer:<\/strong><\/h3>\n\n\n\n
\n\n\n\nExplanation <\/strong><\/h3>\n\n\n\n
hh<\/th> ln\u2061(1+h)h\\frac{\\ln(1 + h)}{h}<\/th><\/tr><\/thead> -0.1<\/td> 0.9486<\/td><\/tr> -0.01<\/td> 0.9950<\/td><\/tr> -0.001<\/td> 0.9995<\/td><\/tr> 0.001<\/td> 1.0005<\/td><\/tr> 0.01<\/td> 1.0050<\/td><\/tr> 0.1<\/td> 1.0486<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
\n\n\n\nWhy Is This Important?<\/strong><\/h3>\n\n\n\n
\n\n\n\nFinal Answer:<\/strong><\/h3>\n\n\n\n
![\"\"](\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/05\/learnexams-banner7-31.jpeg\")
<\/figure>\n","protected":false},"excerpt":{"rendered":"