{"id":40250,"date":"2025-06-27T15:05:28","date_gmt":"2025-06-27T15:05:28","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=40250"},"modified":"2025-06-27T15:05:29","modified_gmt":"2025-06-27T15:05:29","slug":"the-limit-below-represents-derivative-f","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/the-limit-below-represents-derivative-f\/","title":{"rendered":"The limit below represents derivative f"},"content":{"rendered":"\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"602\" height=\"241\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-413.png\" alt=\"\" class=\"wp-image-40251\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-413.png 602w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-413-300x120.png 300w\" sizes=\"auto, (max-width: 602px) 100vw, 602px\" \/><\/figure>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>The limit given is:<\/p>\n\n\n\n<p>lim\u2061h\u21920(4+h)2\u221216h\\lim_{h \\to 0} \\frac{(4 + h)^2 &#8211; 16}{h}<\/p>\n\n\n\n<p>This expression fits the definition of the derivative at a point, which is:<\/p>\n\n\n\n<p>f\u2032(a)=lim\u2061h\u21920f(a+h)\u2212f(a)hf'(a) = \\lim_{h \\to 0} \\frac{f(a + h) &#8211; f(a)}{h}<\/p>\n\n\n\n<p>In the given limit, the numerator is (4+h)2\u221216(4 + h)^2 &#8211; 16. Notice that 16 is 424^2, so we can recognize that f(a+h)=(4+h)2f(a + h) = (4 + h)^2 and f(a)=16f(a) = 16. This means a=4a = 4, and since the function squared the input, we deduce that f(x)=x2f(x) = x^2.<\/p>\n\n\n\n<p><strong>Correct answers:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>f(x)=x2f(x) = x^2<\/li>\n\n\n\n<li>a=4a = 4<\/li>\n<\/ul>\n\n\n\n<p><strong>Explanation<\/strong><\/p>\n\n\n\n<p>To determine the function f(x)f(x) and the value aa from the limit expression, begin by identifying the structure. The given limit has the form used to define the derivative of a function at a point. The general form is lim\u2061h\u21920f(a+h)\u2212f(a)h\\lim_{h \\to 0} \\frac{f(a + h) &#8211; f(a)}{h}, where ff is a function and aa is the specific point of evaluation.<\/p>\n\n\n\n<p>In this expression, f(a+h)f(a + h) appears to be (4+h)2(4 + h)^2, and f(a)f(a) is 16. Observing that 42=164^2 = 16, it becomes evident that a=4a = 4. From f(a+h)=(4+h)2f(a + h) = (4 + h)^2, we infer the general function rule is f(x)=x2f(x) = x^2, since substituting x=4+hx = 4 + h recovers the form f(a+h)f(a + h).<\/p>\n\n\n\n<p>To verify this, apply the function definition and use the difference quotient:<\/p>\n\n\n\n<p>f(4+h)\u2212f(4)h=(4+h)2\u221216h\\frac{f(4 + h) &#8211; f(4)}{h} = \\frac{(4 + h)^2 &#8211; 16}{h}<\/p>\n\n\n\n<p>Expand the square in the numerator:<\/p>\n\n\n\n<p>(4+h)2=16+8h+h2(4 + h)^2 = 16 + 8h + h^2<\/p>\n\n\n\n<p>Subtract 16:<\/p>\n\n\n\n<p>8h+h28h + h^2<\/p>\n\n\n\n<p>Divide the entire expression by hh:<\/p>\n\n\n\n<p>8h+h2h=8+h\\frac{8h + h^2}{h} = 8 + h<\/p>\n\n\n\n<p>Taking the limit as h\u21920h \\to 0, we find:<\/p>\n\n\n\n<p>f\u2032(4)=8f'(4) = 8<\/p>\n\n\n\n<p>This confirms the function and the point of differentiation.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"852\" height=\"1024\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-1247.jpeg\" alt=\"\" class=\"wp-image-40252\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-1247.jpeg 852w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-1247-250x300.jpeg 250w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-1247-768x923.jpeg 768w\" sizes=\"auto, (max-width: 852px) 100vw, 852px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>The Correct Answer and Explanation is: The limit given is: lim\u2061h\u21920(4+h)2\u221216h\\lim_{h \\to 0} \\frac{(4 + h)^2 &#8211; 16}{h} This expression fits the definition of the derivative at a point, which is: f\u2032(a)=lim\u2061h\u21920f(a+h)\u2212f(a)hf'(a) = \\lim_{h \\to 0} \\frac{f(a + h) &#8211; f(a)}{h} In the given limit, the numerator is (4+h)2\u221216(4 + h)^2 &#8211; 16. Notice that [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-40250","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/40250","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=40250"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/40250\/revisions"}],"predecessor-version":[{"id":40253,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/40250\/revisions\/40253"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=40250"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=40250"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=40250"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}