{"id":4053,"date":"2024-01-30T12:30:55","date_gmt":"2024-01-30T06:30:55","guid":{"rendered":"https:\/\/stage-wp.10minuteschool.com\/?p=4053"},"modified":"2024-11-05T14:43:31","modified_gmt":"2024-11-05T08:43:31","slug":"differentiation-limit","status":"publish","type":"post","link":"https:\/\/10minuteschool.com\/content\/differentiation-limit\/","title":{"rendered":"\u09b2\u09bf\u09ae\u09bf\u099f (Limit)"},"content":{"rendered":"

\u09b2\u09bf\u09ae\u09bf\u099f \u0995\u09bf? <\/b>(Limit)\u00a0<\/b><\/span><\/h2>\n

y=f(x)<\/span><\/span> \u098f\u09b0 x <\/span><\/span>\u00a0\u099a\u09b2\u0995\u09c7\u09b0\u09c7 \u09ae\u09be\u09a8 \u098f\u0995\u099f\u09bf \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09a7\u09cd\u09b0\u09c1\u09ac \u09b8\u0982\u0996\u09cd\u09af\u09be <\/span>a<\/span>\u00a0<\/span> \u0985\u09aa\u09c7\u0995\u09cd\u09b7\u09be \u09ac\u09c3\u09b9\u09a4\u09cd\u09a4\u09b0 \u09ac\u09be \u0995\u09cd\u09b7\u09c1\u09a6\u09cd\u09b0\u09a4\u09b0 \u09ae\u09be\u09a8\u0997\u09c1\u09b2\u09bf \u0997\u09cd\u09b0\u09b9\u09a3 \u0995\u09b0\u09c7 \u0995\u09cd\u09b0\u09ae\u09b6 a<\/span> <\/span>\u00a0\u098f\u09b0 \u09a8\u09bf\u0995\u099f\u09ac\u09b0\u09cd\u09a4\u09c0 \u09b9\u0993\u09df\u09be\u09df \u09af\u09a6\u09bf\u00a0 f (x) <\/span> <\/span>\u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09ae\u09be\u09a8 \u0986\u09b0\u09c7\u0995\u099f\u09bf \u09a7\u09cd\u09b0\u09c1\u09ac\u0995 l \u098f\u09b0 \u09a8\u09bf\u0995\u099f\u09ac\u09b0\u09cd\u09a4\u09c0 \u09b9\u09df \u09a4\u09ac\u09c7 1 <\/span> \u0995\u09c7 f (x) <\/span> <\/span>\u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09b8\u09c0\u09ae\u09be\/limit<\/strong><\/a><\/span> \u09ac\u09b2\u09c7<\/span>\u0964<\/span><\/p>\n

\\Rightarrow<\/span> <\/span> \u098f\u0995\u09c7 \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09be \u09b9\u09df l t \/ \\lim <\/span><\/span>\u2061<\/span>\u09a6\u09cd\u09ac\u09be\u09b0\u09be\u0964<\/span><\/p>\n

\\operatorname{Lim} f(x)=l \\rightarrow <\/span><\/span>\u09b8\u09c0\u09ae\u09be\u09b8\u09cd\u09a5\u09ae\u09be\u09a8<\/span>\u0964<\/span><\/p>\n

\u09af\u09c7\u09ae\u09a8\u0983\u00a0<\/span><\/p>\n

f(x)<\/span>=\\frac{x^{2}-4}{x-2} <\/span><\/span><\/p>\n

\\frac{\\lim }{x \\rightarrow 0} f(x) <\/span> =\u00a0 \\frac{\\lim }{x \\rightarrow 0} \\frac{x^{2}-4}{x-2} <\/span> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <\/span>\u2061<\/span><\/p>\n\n\n\n\n\n\n\n\n
x <\/span><\/span><\/td>\nf(x) <\/span><\/span><\/td>\n<\/tr>\n
2<\/span><\/td>\n\\alpha <\/span><\/td>\n<\/tr>\n
1.9<\/span><\/td>\n3.9<\/span><\/td>\n<\/tr>\n
1.9999<\/span><\/td>\n3.9999<\/span><\/td>\n<\/tr>\n
2.01<\/span><\/td>\n4.01<\/span><\/td>\n<\/tr>\n
2.000001<\/span><\/td>\n4.000001<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

\"\u09b2\u09bf\u09ae\u09bf\u099f\"<\/p>\n

<\/h3>\n

\u27f9<\/b> \u09ac\u09be\u09ae\u09a6\u09bf\u0995\u09ac\u09b0\u09cd\u09a4\u09c0 \u09b2\u09bf\u09ae\u09bf\u099f <\/b>(The limit on the left) :<\/b><\/span><\/h3>\n

x <\/span> \u099a\u09b2\u0995 <\/span> a <\/span> \u0985\u09aa\u09c7\u0995\u09cd\u09b7\u09be\u09df \u0995\u09cd\u09b7\u09c1\u09a6\u09cd\u09b0\u09a4\u09ae \u09ae\u09be\u09a8 \u0997\u09cd\u09b0\u09b9\u09a8 \u0995\u09b0\u09c7 \u09ac\u09be\u09ae\u09a6\u09bf\u0995 \u09b9\u09a4\u09c7 \u0995\u09cd\u09b0\u09ae\u09b6 <\/span> a <\/span>\u098f\u09b0 \u09a8\u09bf\u0995\u099f\u09ac\u09b0\u09cd\u09a4\u09c0 \u09b9\u0993\u09df\u09be\u09df \u09af\u09a6\u09bf <\/span> f (x)<\/span><\/span> \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09ae\u09be\u09a8 \u09a7\u09cd\u09b0\u09c1\u09ac\u0995 1 <\/span><\/span>\u098f\u09b0 \u09a8\u09bf\u0995\u099f\u09ac\u09b0\u09cd\u09a4\u09c0 \u09b9\u09df \u09a4\u09ac\u09c7 1 <\/span><\/span> \u0995\u09c7 <\/span> f (x)\u00a0 <\/span><\/span>\u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09ac\u09be\u09ae\u09a6\u09bf\u0995\u09ac\u09b0\u09cd\u09a4\u09c0 \u09ac\u09be \u09ac\u09be\u09ae\u09be\u09ac\u09b0\u09cd\u09a4\u09c0 \u09b2\u09bf\u09ae\u09bf\u099f<\/a><\/span> \u09ac\u09b2\u09c7\u0964<\/span><\/p>\n

\\frac{\\operatorname{Lim}}{x \\rightarrow a^{-}} f(x) <\/span>=l <\/span><\/span><\/p>\n

\u27f9<\/b> \u09a1\u09be\u09a8\u09a6\u09bf\u0995\u09ac\u09b0\u09cd\u09a4\u09c0 \u09b2\u09bf\u09ae\u09bf\u099f <\/b>(The limit on the right) :<\/b><\/span><\/h3>\n

x <\/span> \u099a\u09b2\u0995 a <\/span><\/span>\u0985\u09aa\u09c7\u0995\u09cd\u09b7\u09be\u09df \u09ac\u09c3\u09b9\u09a4\u09cd\u09a4\u09ae \u09ae\u09be\u09a8\u0997\u09c1\u09b2\u09bf \u0997\u09cd\u09b0\u09b9\u09a8 \u0995\u09b0\u09c7 \u09a1\u09be\u09a8\u09a6\u09bf\u0995 \u09b9\u09a4\u09c7 \u0995\u09cd\u09b0\u09ae\u09b6 a <\/span><\/span>‘<\/span> \u098f\u09b0 \u09a8\u09bf\u0995\u099f\u09ac\u09b0\u09cd\u09a4\u09c0 \u09b9\u0993\u09df\u09be\u09df \u09af\u09a6\u09bf f (x) <\/span><\/span> \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09ae\u09be\u09a8 \u09a7\u09cd\u09b0\u09c1\u09ac\u0995 1 <\/span>\u098f\u09b0 \u09a8\u09bf\u0995\u099f\u09ac\u09b0\u09cd\u09a4\u09c0 \u09b9\u09df \u09a4\u09ac\u09c7 1 <\/span>\u0995\u09c7 f (x) <\/span><\/span>\u00a0<\/span>\u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09a1\u09be\u09a8\u09a6\u09bf\u0995\u09ac\u09b0\u09cd\u09a4\u09c0 \u09ac\u09be \u09a1\u09be\u09a8\u09be\u09ac\u09b0\u09cd\u09a4\u09c0 \u09b2\u09bf\u09ae\u09bf\u099f \u09ac\u09b2\u09c7\u0964<\/span><\/p>\n

\\frac{\\operatorname{Lim}}{x \\rightarrow a^{+}} f(x) <\/span>=l <\/span><\/span><\/p>\n

\u27f9<\/b> \u09b8\u09c0\u09ae\u09be\u09b0 \u09ae\u09cc\u09b2\u09bf\u0995 \u09a7\u09b0\u09cd\u09ae\u09be\u09ac\u09b2\u09c0 <\/b>(The basic religions of the range):<\/b><\/span><\/h3>\n

\\frac{\\operatorname{lim}}{x \\rightarrow a} f(x)=<\/span> l<\/span>\u00a0 \u00a0\u098f\u09ac\u0982 \\frac{\\operatorname{Lim}}{x \\rightarrow a} g(x)<\/span>=m <\/span><\/span><\/p>\n

\u00a01. \\frac{\\operatorname{lim}}{x \\rightarrow a}\\{f(x) \\pm g(x)\\} <\/span>=\\frac{\\operatorname{Lim}}{x \\rightarrow a} f(x) \\pm \\frac{\\operatorname{Lim}}{x \\rightarrow a} g(x) j <\/span>=l \\pm m <\/span><\/span><\/p>\n

\u00a02.\\frac{\\operatorname{lim}}{x \\rightarrow a} c f(x) <\/span>=c \\frac{\\operatorname{Lim}}{x \\rightarrow a} f(x) <\/span>; c=<\/span>\u09a7\u09cd\u09b0\u09c1\u09ac\u0995<\/p>\n

3. \\frac{\\operatorname{lim}}{x \\rightarrow a} f(x) g(x) <\/span>=\\frac{\\operatorname{Lim}}{x \\rightarrow a} f(x) \\cdot \\frac{\\operatorname{Lim}}{x \\rightarrow a} g(x) <\/span>=\\operatorname{lm} <\/span><\/span><\/p>\n

4.\\frac{\\lim }{x \\rightarrow a} \\frac{1}{f(x)} <\/span>=\\frac{1}{l} <\/span>; if l \\neq 0 <\/span><\/span><\/p>\n

5. \\frac{\\operatorname{lim}}{x \\rightarrow a} \\frac{f(x)}{g(x)}=\\frac{l}{m} ;<\/span> if m \\neq 0<\/span><\/p>\n

6. \\frac{\\operatorname{lim}}{x \\rightarrow a} c=c ; \\mathrm{c}= \\text {\u09a7\u09cd\u09b0\u09c1\u09ac\u0995}<\/span><\/p>\n

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\u098f\u0987\u099a\u098f\u09b8\u09b8\u09bf \u0993 \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u09aa\u09b0\u09c0\u0995\u09cd\u09b7\u09be\u09b0\u09cd\u09a5\u09c0\u09a6\u09c7\u09b0 \u099c\u09a8\u09cd\u09af \u0986\u09ae\u09be\u09a6\u09c7\u09b0 \u0995\u09cb\u09b0\u09cd\u09b8\u09b8\u09ae\u09c2\u09b9\u0983<\/strong><\/em><\/p>\n<\/div>\n<\/div>\n<\/div>\n