{"id":40239,"date":"2025-06-27T14:53:02","date_gmt":"2025-06-27T14:53:02","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=40239"},"modified":"2025-06-27T14:53:03","modified_gmt":"2025-06-27T14:53:03","slug":"the-limit-below-represents-a-derivative-f-2","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/the-limit-below-represents-a-derivative-f-2\/","title":{"rendered":"The limit below represents a derivative f&#8217;"},"content":{"rendered":"\n<p>The limit below represents a derivative f'(a). Find f(x) and a. lim h->0 = 6^3+h &#8211; 216 \/ h f(x)= a=<\/p>\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>Let&#8217;s carefully analyze the given limit:lim\u2061h\u2192063+h\u2212216h\\lim_{h \\to 0} \\frac{6^{3 + h} &#8211; 216}{h}h\u21920lim\u200bh63+h\u2212216\u200b<\/p>\n\n\n\n<p>This expression resembles the definition of a derivative, which is generally written as:f\u2032(a)=lim\u2061h\u21920f(a+h)\u2212f(a)hf'(a) = \\lim_{h \\to 0} \\frac{f(a + h) &#8211; f(a)}{h}f\u2032(a)=h\u21920lim\u200bhf(a+h)\u2212f(a)\u200b<\/p>\n\n\n\n<p>Comparing this with the provided limit:63+h\u2212216h\\frac{6^{3 + h} &#8211; 216}{h}h63+h\u2212216\u200b<\/p>\n\n\n\n<p>It means:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>f(a+h)=63+hf(a + h) = 6^{3 + h}f(a+h)=63+h<\/li>\n\n\n\n<li>f(a)=216f(a) = 216f(a)=216<\/li>\n<\/ul>\n\n\n\n<p>First, find f(x)f(x)f(x).<\/p>\n\n\n\n<p>Looking at f(a+h)=63+hf(a + h) = 6^{3 + h}f(a+h)=63+h, this suggests f(x)=6xf(x) = 6^{x}f(x)=6x.<\/p>\n\n\n\n<p>To confirm, let&#8217;s compute:<\/p>\n\n\n\n<p>If f(x)=6xf(x) = 6^{x}f(x)=6x, then:f(a+h)=6a+hf(a + h) = 6^{a + h}f(a+h)=6a+hf(a)=6a=216f(a) = 6^{a} = 216f(a)=6a=216<\/p>\n\n\n\n<p>We are told f(a)=216f(a) = 216f(a)=216, so:6a=2166^{a} = 2166a=216<\/p>\n\n\n\n<p>We know that:63=2166^{3} = 21663=216<\/p>\n\n\n\n<p>Thus:a=3a = 3a=3<\/p>\n\n\n\n<p>Therefore:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The function is f(x)=6xf(x) = 6^{x}f(x)=6x<\/li>\n\n\n\n<li>The value of a=3a = 3a=3<\/li>\n<\/ul>\n\n\n\n<p><strong>Final Answer:<\/strong>f(x)=6xanda=3f(x) = 6^{x} \\quad \\text{and} \\quad a = 3f(x)=6xanda=3<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation<\/h3>\n\n\n\n<p>The given limit represents the definition of a derivative at a specific point. The general formula for the derivative of a function f(x)f(x)f(x) at x=ax = ax=a 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}f\u2032(a)=h\u21920lim\u200bhf(a+h)\u2212f(a)\u200b<\/p>\n\n\n\n<p>In the provided expression:lim\u2061h\u2192063+h\u2212216h\\lim_{h \\to 0} \\frac{6^{3 + h} &#8211; 216}{h}h\u21920lim\u200bh63+h\u2212216\u200b<\/p>\n\n\n\n<p>We can match this with the derivative formula by recognizing f(a+h)=63+hf(a + h) = 6^{3 + h}f(a+h)=63+h and f(a)=216f(a) = 216f(a)=216. It implies that the original function f(x)f(x)f(x) must be f(x)=6xf(x) = 6^{x}f(x)=6x.<\/p>\n\n\n\n<p>To verify, substitute x=ax = ax=a into the function:f(a)=6a=216f(a) = 6^{a} = 216f(a)=6a=216<\/p>\n\n\n\n<p>We are asked to find the value of aaa such that 6a=2166^{a} = 2166a=216. Knowing that:63=6\u00d76\u00d76=36\u00d76=2166^{3} = 6 \\times 6 \\times 6 = 36 \\times 6 = 21663=6\u00d76\u00d76=36\u00d76=216<\/p>\n\n\n\n<p>So a=3a = 3a=3.<\/p>\n\n\n\n<p>Therefore, the function f(x)=6xf(x) = 6^{x}f(x)=6x and the value a=3a = 3a=3 satisfy the given limit expression. This approach demonstrates how to connect a limit expression with the derivative definition to extract both the function and the evaluation point.<\/p>\n\n\n\n<p>The correct function and value are:f(x)=6x,a=3f(x) = 6^{x}, \\quad a = 3f(x)=6x,a=3<\/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-banner8-1244.jpeg\" alt=\"\" class=\"wp-image-40240\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-1244.jpeg 852w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-1244-250x300.jpeg 250w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-1244-768x923.jpeg 768w\" sizes=\"auto, (max-width: 852px) 100vw, 852px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>The limit below represents a derivative f'(a). Find f(x) and a. lim h->0 = 6^3+h &#8211; 216 \/ h f(x)= a= The Correct Answer and Explanation is: Let&#8217;s carefully analyze the given limit:lim\u2061h\u2192063+h\u2212216h\\lim_{h \\to 0} \\frac{6^{3 + h} &#8211; 216}{h}h\u21920lim\u200bh63+h\u2212216\u200b This expression resembles the definition of a derivative, which is generally written as:f\u2032(a)=lim\u2061h\u21920f(a+h)\u2212f(a)hf'(a) = \\lim_{h [&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-40239","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/40239","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=40239"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/40239\/revisions"}],"predecessor-version":[{"id":40241,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/40239\/revisions\/40241"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=40239"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=40239"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=40239"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}