帮我解决几道大学高数题
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帮我解决几道大学高数题
lim(x_0)(e^ax -1)/sinbx
lim(x-1)(x-1)/(x-π)
lim(1-1/x)^根号x
lim(x-无穷){(2x+1)/(2x-1 )}^x
lim(x-无穷)(1+2^x+3^x)^1/x
会几道就帮我解几道吧,
lim(x_0)(e^ax -1)/sinbx
lim(x-1)(x-1)/(x-π)
lim(1-1/x)^根号x
lim(x-无穷){(2x+1)/(2x-1 )}^x
lim(x-无穷)(1+2^x+3^x)^1/x
会几道就帮我解几道吧,
![帮我解决几道大学高数题](/uploads/image/z/15158117-29-7.jpg?t=%E5%B8%AE%E6%88%91%E8%A7%A3%E5%86%B3%E5%87%A0%E9%81%93%E5%A4%A7%E5%AD%A6%E9%AB%98%E6%95%B0%E9%A2%98)
1.lim(x->0) (ax)/(bx) = a/b
2.lim(x->1) (x-1)/(x-π) = 0
3.lim(x->+∞) (1-1/x )^√x = lim(x->+∞) (1-1/x )^x * (1/√x)
lim(x->+∞) (1-1/x )^x = e^(-1) = 1/e,1/√x -> 0
原式 = 1
4.lim(x->∞) [1+ 1/(x+1/2) ] ^ (x+1/2) * [ x /(x+1/2)]
= e^1 = e
5.3^x < 1+2^x+3^x < 3 * 3^x
3 < (1+2^x+3^x)^1/x < 3 ^(1/x) * 3
lim(x->∞) 3 ^(1/x) = 3^0 = 1 由迫敛准则(夹挤准则),得:
lim(x->∞) (1+2^x+3^x)^1/x = 3
再问: 谢谢。能不能再帮我看看这个 lim(x-0)(x^2 sin1/x)/(根号2x^2-1) lim(x-1)(绝对值x-1)/x-1 已知lim(x趋于-1)(2x^2+ax+b)/(x+3)=3 求a、b 设X0=1 X1+1+X0/1+X0...Xn+1=1+Xn/1+Xn 证明lim(x趋于无穷)X存在,并求之。 X后面的数字是下标
再答: 1. lim(x->0)(x^2 sin1/x)/(根号2x^2-1) = 0 无穷小与有界函数的乘积还是无穷小 2. lim(x->1) | x-1| /(x-1) 不存在, 因为左极限 = -1, 而右极限 =1 3. lim(x->-3)(2x^2+ax+b) / (x+3) = 3 => 2 * 3^2 - 3a + b =0 => a = 6 + b/3 => 2x^2+ax+b = 2(x+3)(x+b/6) lim(x->-3)(2x^2+ax+b) / (x+3) = lim(x->-3)(2(x+3)(x+b/6) / (x+3) = lim(x->-3) 2(x+b/6) = 3 => b = 27, a = 15 4. X(n+1)=1 + X(n) / [1+X(n)] 可以用数学归纳法证明 {X(n)} 单增且 1 < X(n) < 2 由单调有界准则知:{X(n)} 收敛, 设其极限为A, X(n+1)=1 + X(n) / [1+X(n)] =》 A = 1 + A/(1+A) => A = (1+√5) /2
2.lim(x->1) (x-1)/(x-π) = 0
3.lim(x->+∞) (1-1/x )^√x = lim(x->+∞) (1-1/x )^x * (1/√x)
lim(x->+∞) (1-1/x )^x = e^(-1) = 1/e,1/√x -> 0
原式 = 1
4.lim(x->∞) [1+ 1/(x+1/2) ] ^ (x+1/2) * [ x /(x+1/2)]
= e^1 = e
5.3^x < 1+2^x+3^x < 3 * 3^x
3 < (1+2^x+3^x)^1/x < 3 ^(1/x) * 3
lim(x->∞) 3 ^(1/x) = 3^0 = 1 由迫敛准则(夹挤准则),得:
lim(x->∞) (1+2^x+3^x)^1/x = 3
再问: 谢谢。能不能再帮我看看这个 lim(x-0)(x^2 sin1/x)/(根号2x^2-1) lim(x-1)(绝对值x-1)/x-1 已知lim(x趋于-1)(2x^2+ax+b)/(x+3)=3 求a、b 设X0=1 X1+1+X0/1+X0...Xn+1=1+Xn/1+Xn 证明lim(x趋于无穷)X存在,并求之。 X后面的数字是下标
再答: 1. lim(x->0)(x^2 sin1/x)/(根号2x^2-1) = 0 无穷小与有界函数的乘积还是无穷小 2. lim(x->1) | x-1| /(x-1) 不存在, 因为左极限 = -1, 而右极限 =1 3. lim(x->-3)(2x^2+ax+b) / (x+3) = 3 => 2 * 3^2 - 3a + b =0 => a = 6 + b/3 => 2x^2+ax+b = 2(x+3)(x+b/6) lim(x->-3)(2x^2+ax+b) / (x+3) = lim(x->-3)(2(x+3)(x+b/6) / (x+3) = lim(x->-3) 2(x+b/6) = 3 => b = 27, a = 15 4. X(n+1)=1 + X(n) / [1+X(n)] 可以用数学归纳法证明 {X(n)} 单增且 1 < X(n) < 2 由单调有界准则知:{X(n)} 收敛, 设其极限为A, X(n+1)=1 + X(n) / [1+X(n)] =》 A = 1 + A/(1+A) => A = (1+√5) /2