{"id":6009,"date":"2021-10-04T22:18:36","date_gmt":"2021-10-04T16:48:36","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6009"},"modified":"2021-10-04T22:19:18","modified_gmt":"2021-10-04T16:49:18","slug":"limits-examples","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/limits-examples\/","title":{"rendered":"Limits Examples"},"content":{"rendered":"

Here you will learn some limits examples for better understanding of limit concepts<\/a><\/span>.<\/p>\n\n\n

\n
\n\n

Example 1 : <\/span>If \\(\\displaystyle{\\lim_{x \\to \\infty}}\\)(\\({x^3+1\\over x^2+1}-(ax+b)\\)) = 2, then find the value of a and b.<\/p>\n

Solution : <\/span>\t\\(\\displaystyle{\\lim_{x \\to \\infty}}\\)(\\({x^3+1\\over x^2+1}-(ax+b)\\)) = 2

\\(\\implies\\) \\(\\displaystyle{\\lim_{x \\to \\infty}}\\)\\(x^3(1-a)-bx^2-ax+(1-b)\\over x^2+1\\) = 2

\\(\\implies\\) \\(\\displaystyle{\\lim_{x \\to \\infty}}\\)\\(x(1-a)-b-{a\\over x}+{(1-b)\\over x^2}\\over 1+{1\\over x^2}\\) = 2

\\(\\implies\\) 1 – a = 0, -b = 2 \\(\\implies\\) a = 1, b = -2<\/p>\n


\n\n \n

Example 2 : <\/span> Evaluate : \\(\\displaystyle{\\lim_{x \\to 0}}\\) \\((2+x)sin(2+x)-2sin2\\over x\\)<\/p>\n

Solution : <\/span>\\(\\displaystyle{\\lim_{x \\to 0}}\\) \\(2(sin(2+x)-sin2)+xsin(2+x)\\over x\\)

\n = \\(\\displaystyle{\\lim_{x \\to 0}}\\)(\\(2.2.cos(2+{x\\over 2})sin{x\\over 2}\\over x\\) + sin(2+x))

= \\(\\displaystyle{\\lim_{x \\to 0}}\\)\\(2cos(2+{x\\over 2})sin{x\\over 2}\\over {x\\over 2}\\) + \\(\\displaystyle{\\lim_{x \\to 0}}\\)sin(2+x)

= 2cos2 + sin2\n <\/p>


\n\n \n

Example 3 : <\/span>Evaluate : \\(\\displaystyle{\\lim_{x \\to 0}}\\) \\(xln(1+2tanx)\\over 1-cosx\\)<\/p>\n

Solution : <\/span>\\(\\displaystyle{\\lim_{x \\to 0}}\\) \\(xln(1+2tanx)\\over 1-cosx\\)

= \\(\\displaystyle{\\lim_{x \\to 0}}\\) \\(xln(1+2tanx)\\over {1-cosx\\over x^2}.x^2\\).\\(2tanx\\over 2tanx\\)

= 4<\/p>\n


\n\n

Example 4 : <\/span>Evaluate : \\(\\displaystyle{\\lim_{x \\to \\infty}}\\) \\(({7x^2+1\\over 5x^2-1})^{x^5\\over {1-x^3}}\\)<\/p>\n

Solution : <\/span>Here f(x) = \\({7x^2+1\\over 5x^2-1}\\)

\\(\\phi\\)(x) = \\({x^5\\over {1-x^3}}\\) = \\(x^2x^3\\over 1-x^3\\) = \\(x^2\\over {1\\over x^3}-1\\)

\\(\\therefore\\)     \\(\\displaystyle{\\lim_{x \\to \\infty}}\\) f(x) = \\(7\\over 5\\)    &     \\(\\displaystyle{\\lim_{x \\to \\infty}}\\) \\(\\phi\\)(x) \\(\\rightarrow\\) – \\(\\infty\\)

\\(\\implies\\) \\(\\displaystyle{\\lim_{x \\to \\infty}}\\) \\((f(x))^{\\phi (x)}\\) = \\(({7\\over 5})^{-\\infty}\\) = 0<\/p>\n
\n\n

Practice these given limits examples to test your knowledge on concepts of limits.<\/p> \n <\/div>\n <\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn some limits examples for better understanding of limit concepts. Example 1 : If \\(\\displaystyle{\\lim_{x \\to \\infty}}\\)(\\({x^3+1\\over x^2+1}-(ax+b)\\)) = 2, then find the value of a and b. Solution : \\(\\displaystyle{\\lim_{x \\to \\infty}}\\)(\\({x^3+1\\over x^2+1}-(ax+b)\\)) = 2 \\(\\implies\\) \\(\\displaystyle{\\lim_{x \\to \\infty}}\\)\\(x^3(1-a)-bx^2-ax+(1-b)\\over x^2+1\\) = 2 \\(\\implies\\) \\(\\displaystyle{\\lim_{x \\to \\infty}}\\)\\(x(1-a)-b-{a\\over x}+{(1-b)\\over x^2}\\over 1+{1\\over x^2}\\) = …<\/p>\n

Limits Examples<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","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":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[19],"tags":[],"yoast_head":"\nLimits Examples - Mathemerize<\/title>\n<meta name=\"description\" content=\"Solve these Limits examples to practice and test your knowledge of Limits concepts.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/limits-examples\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Limits Examples - 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