$$lim_{x\\rightarrow 0}(1+x)^n ~ 1+nx$$<\/p>\n\n\n\n
Proof:<\/p>\n\n\n\n
Taylor Expansion at x=0<\/p>\n\n\n\n
$$ (1+x)^n\\approx (1+x)^n|_{x=0} + \\frac{d (1+x)^n}{d x}|_{x=0} \\times (x-0) + \\frac{d^2 (1+x)^n}{d x^2}|_{x=0}\\times (x-0)^2 + O(x^3) $$<\/p>\n\n\n\n
$$ \\approx 1 + nx + \\frac{n(n-1)}{2} x^2 + O(x^3)$$<\/p>\n\n\n\n
$$ \\approx 1 + nx + O(x^2)$$<\/p>\n","protected":false},"excerpt":{"rendered":"
$$lim_{x\\rightarrow 0}(1+x)^n ~ 1+nx$$ Proof: Taylor Expansion at x=0 $$ (1+x)^n\\approx (1+x)^n|_{x=0} + \\frac{d (1+x)^n}{d x}|_{x=0} \\times (x-0) + \\frac{d^2 (1+x)^n}{d x^2}|_{x=0}\\times (x-0)^2 + O(x^3) $$ $$ \\approx 1 + nx + \\frac{n(n-1)}{2} x^2 + O(x^3)$$ $$ \\approx 1 + nx + O(x^2)$$<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6,18],"tags":[],"_links":{"self":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/3367"}],"collection":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=3367"}],"version-history":[{"count":6,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/3367\/revisions"}],"predecessor-version":[{"id":3373,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/3367\/revisions\/3373"}],"wp:attachment":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3367"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3367"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3367"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}