Find the limit: lim_{xrightarrowinfty}frac{3x^6-x^4+2x^2-3}{4x^5+2x^3-5}

Find the limit: lim_{xrightarrowinfty}frac{3x^6-x^4+2x^2-3}{4x^5+2x^3-5}

Question
Limits and continuity
asked 2021-01-02
Find the limit:
\(\lim_{x\rightarrow\infty}\frac{3x^6-x^4+2x^2-3}{4x^5+2x^3-5}\)

Answers (1)

2021-01-03
\(\lim_{x\rightarrow\infty}\frac{3x^6-x^4+2x^2-3}{4x^5+2x^3-5}\)
Multiply and divided by \(x^5\):
\(\lim_{x\rightarrow\infty}\frac{3x^6-x^4+2x^2-3}{4x^5+2x^3-5}=\lim_{x\rightarrow\infty}\frac{x^5\frac{3x^6-x^4+2x^2-3}{x^5}}{x^5\frac{4x^5+2x^3-5}{x^5}}\)
Divide:
\(\lim_{x\rightarrow\infty}\frac{x^5\frac{3x^6-x^4+2x^2-3}{x^5}}{x^5\frac{4x^5+2x^3-5}{x^5}}=\lim_{x\rightarrow\infty}\frac{3x-\frac{1}{x}+\frac{2}{x^3}-\frac{3}{x^5}}{4+\frac{2}{x^2}-\frac{5}{x^5}}\)
The limit of the quotient is the quotient of limits:
\(\lim_{x\rightarrow\infty}\frac{3x-\frac{1}{x}+\frac{2}{x^3}-\frac{3}{x^5}}{4+\frac{2}{x^2}-\frac{5}{x^5}}=\frac{\lim_{x\rightarrow\infty}(3x-\frac{1}{x}+\frac{2}{x^3}-\frac{3}{x^5})}{\lim_{x\rightarrow\infty}(4+\frac{2}{x^2}-\frac{5}{x^5})}\)
The limit of a sum/difference is the sum/difference of limits: \(\frac{\lim_{x\rightarrow\infty}(3x-\frac{1}{x}+\frac{2}{x^3}-\frac{3}{x^5})}{\lim_{x\rightarrow\infty}(4+\frac{2}{x^2}-\frac{5}{x^5})}=\frac{(-\lim_{x\rightarrow\infty}\frac{3}{x^5}+\lim_{x\rightarrow\infty}\frac{2}{x^3}-\lim_{x\rightarrow\infty}\frac{1}{x}+\lim_{x\rightarrow\infty}3x)}{\lim_{x\rightarrow\infty}(4+\frac{2}{x^2}-\frac{5}{x^5})}\)
The limit of a quotient is the quotient of limits:
\(\frac{(-\lim_{x\rightarrow\infty}\frac{3}{x^5}+\lim_{x\rightarrow\infty}\frac{2}{x^3}-\lim_{x\rightarrow\infty}\frac{1}{x}+\lim_{x\rightarrow\infty}3x)}{\lim_{x\rightarrow\infty}(4+\frac{2}{x^2}-\frac{5}{x^5})}=\frac{(-\lim_{x\rightarrow\infty}\frac{3}{x^5}+\lim_{x\rightarrow\infty}\frac{2}{x^3}-\lim_{x\rightarrow\infty}3x)-\frac{\lim_{x\rightarrow\infty}1}{\lim_{x\rightarrow\infty}x}}{\lim_{x\rightarrow\infty}(4+\frac{2}{x^2}-\frac{5}{x^5})} \)
The limit of a constant is equal to the constant:
\(\frac{-\lim_{x\rightarrow\infty}\frac{3}{x^5}+\lim_{x\rightarrow\infty}\frac{2}{x^3}-\lim_{x\rightarrow\infty}3x-\frac{\lim_{x\rightarrow\infty}1}{\lim_{x\rightarrow\infty}x}}{\lim_{x\rightarrow\infty}(4+\frac{2}{x^2}-\frac{5}{x^5})}=\frac{-\lim_{x\rightarrow\infty}\frac{3}{x^5}+\lim_{x\rightarrow\infty}\frac{2}{x^3}-\lim_{x\rightarrow\infty}3x-\frac{1}{\lim_{x\rightarrow\infty}x}}{\lim_{x\rightarrow\infty}(4+\frac{2}{x^2}-\frac{5}{x^5})}\)
Constant divided by a very big number equals 0:
\(\frac{-\lim_{x\rightarrow\infty}\frac{3}{x^5}+\lim_{x\rightarrow\infty}\frac{2}{x^3}-\lim_{x\rightarrow\infty}3x-\frac{1}{\lim_{x\rightarrow\infty}x}}{\lim_{x\rightarrow\infty}(4+\frac{2}{x^2}-\frac{5}{x^5})}=\frac{-\lim_{x\rightarrow\infty}\frac{3}{x^5}+\lim_{x\rightarrow\infty}\frac{2}{x^3}-\lim_{x\rightarrow\infty}3x-(0)}{\lim_{x\rightarrow\infty}(4+\frac{2}{x^2}-\frac{5}{x^5})}\)
Apply the constant multiple rule \(\lim_{x\rightarrow\infty}cf(x)=c\cdot\lim_{x\rightarrow\infty}f(x)\ with\ c=3\ and\ f(x)=\frac{1}{x^5}\)
\(\frac{-\lim_{x\rightarrow\infty}\frac{3}{x^5}+\lim_{x\rightarrow\infty}\frac{2}{x^3}-\lim_{x\rightarrow\infty}3x}{\lim_{x\rightarrow\infty}(4+\frac{2}{x^2}-\frac{5}{x^5})}=\frac{\lim_{x\rightarrow\infty}\frac{2}{x^3}-\lim_{x\rightarrow\infty}3x-(3\lim_{x\rightarrow\infty}\frac{1}{x^5})}{\lim_{x\rightarrow\infty}(4+\frac{2}{x^2}-\frac{5}{x^5})}\)
The limit of a quotient is the quotient of limits:
\(\frac{\lim_{x\rightarrow\infty}\frac{2}{x^3}-\lim_{x\rightarrow\infty}3x-3\lim_{x\rightarrow\infty}\frac{1}{x^5}}{\lim_{x\rightarrow\infty}(4+\frac{2}{x^2}-\frac{5}{x^5})}=\frac{\lim_{x\rightarrow\infty}\frac{2}{x^3}-\lim_{x\rightarrow\infty}3x-3\frac{\lim_{x\rightarrow\infty}1}{\lim_{x\rightarrow\infty}x^5}}{\lim_{x\rightarrow\infty}(4+\frac{2}{x^2}-\frac{5}{x^5})}\)
The limit of a constant is equal to the constant:
\(\frac{\lim_{x\rightarrow\infty}\frac{2}{x^3}-\lim_{x\rightarrow\infty}3x-3\frac{\lim_{x\rightarrow\infty}1}{\lim_{x\rightarrow\infty}x^5}}{\lim_{x\rightarrow\infty}(4+\frac{2}{x^2}-\frac{5}{x^5})}=\frac{\lim_{x\rightarrow\infty}\frac{2}{x^3}-\lim_{x\rightarrow\infty}3x-\frac{3}{\lim_{x\rightarrow\infty}x^5}}{\lim_{x\rightarrow\infty}(4+\frac{2}{x^2}-\frac{5}{x^5})}\)
Constant divided by a very big number equals 0:
\(\frac{\lim_{x\rightarrow\infty}\frac{2}{x^3}-\lim_{x\rightarrow\infty}3x-\frac{3}{\lim_{x\rightarrow\infty}x^5}}{\lim_{x\rightarrow\infty}(4+\frac{2}{x^2}-\frac{5}{x^5})}=\frac{\lim_{x\rightarrow\infty}\frac{2}{x^3}-\lim_{x\rightarrow\infty}3x-3(0)}{\lim_{x\rightarrow\infty}(4+\frac{2}{x^2}-\frac{5}{x^5})}\)
Apply the constant multiple rule:
\(\frac{\lim_{x\rightarrow\infty}\frac{2}{x^3}-\lim_{x\rightarrow\infty}3x}{\lim_{x\rightarrow\infty}(4+\frac{2}{x^2}-\frac{5}{x^5})}=\frac{\lim_{x\rightarrow\infty}3x+(2\lim_{x\rightarrow\infty}\frac{1}{x^3})}{\lim_{x\rightarrow\infty}(4+\frac{2}{x^2}-\frac{5}{x^5})}\)
The limit of a quotient is the quotient of limits:
\(\frac{\lim_{x\rightarrow\infty}3x+(2\lim_{x\rightarrow\infty}\frac{1}{x^3})}{\lim_{x\rightarrow\infty}(4+\frac{2}{x^2}-\frac{5}{x^5})}=\frac{\lim_{x\rightarrow\infty}3x+2\frac{\lim_{x\rightarrow\infty}1}{\lim_{x\rightarrow}x^3}}{\lim_{x\rightarrow\infty}(4+\frac{2}{x^2}-\frac{5}{x^5})}\)
The limit of constant is equal to the constant:
\(\frac{\lim_{x\rightarrow\infty}3x+2\frac{\lim_{x\rightarrow\infty}1}{\lim_{x\rightarrow}x^3}}{\lim_{x\rightarrow\infty}(4+\frac{2}{x^2}-\frac{5}{x^5})}=\frac{\lim_{x\rightarrow\infty}3x+\frac{2}{\lim_{x\rightarrow}x^3}}{\lim_{x\rightarrow\infty}(4+\frac{2}{x^2}-\frac{5}{x^5})}\)
Constant divided by a very big number equals 0:
\(\frac{\lim_{x\rightarrow\infty}3x+\frac{2}{\lim_{x\rightarrow}x^3}}{\lim_{x\rightarrow\infty}(4+\frac{2}{x^2}-\frac{5}{x^5})}=\frac{\lim_{x\rightarrow\infty}3x+0}{\lim_{x\rightarrow\infty}(4+\frac{2}{x^2}-\frac{5}{x^5})}\)
Apply the constant multiple rule:
\(\frac{\lim_{x\rightarrow\infty}3x}{\lim_{x\rightarrow\infty}(4+\frac{2}{x^2}-\frac{5}{x^5})}=\frac{3\lim_{x\rightarrow\infty}x}{\lim_{x\rightarrow\infty}(4+\frac{2}{x^2}-\frac{5}{x^5})}\)
The function grows without a bound:
\(\lim_{x\rightarrow\infty}x=\infty\)
The limit of a sum/difference is the sum/difference of limits:
\(\infty\lim_{x\rightarrow\infty}(4+\frac{2}{x^2}-\frac{5}{x^5})^{-1}\)
\(=\infty(\lim_{x\rightarrow\infty}4+\lim_{x\rightarrow\infty}\frac{2}{x^2}-\lim_{x\rightarrow\infty}\frac{5}{x^5})^{-1}\)
Apply the constant multiple rule
\(\infty(4+\lim_{x\rightarrow\infty}\frac{2}{x^2}-\lim_{x\rightarrow\infty}\frac{5}{x^5})^{-1}\)
\(\infty(4+\lim_{x\rightarrow\infty}\frac{2}{x^2}-5\lim_{x\rightarrow\infty}\frac{1}{x^5})^{-1}\)
The limit of a qutient is the quotient of limits:
\(=\infty(4+\lim_{x\rightarrow\infty}\frac{2}{x^2}-5\frac{\lim_{x\rightarrow\infty}1}{\lim_{x\rightarrow\infty}x^5})^{-1}\)
The limit of a constant is equal to the constant:
\(=\infty(4+\lim_{x\rightarrow\infty}\frac{2}{x^2}-\frac{5}{\lim_{x\rightarrow\infty}x^5})^{-1}\)
Constant divided by a very big number equals 0:
\(=\infty(4+\lim_{x\rightarrow\infty}\frac{2}{x^2}-5(0))^{-1}\)
Apply the constant multiple rule:
\(=\infty(4+2\lim_{x\rightarrow\infty}\frac{1}{x^2})^{-1}\)
The limit of a constant is equal to the constant:
\(=\infty(4+2(0))^{-1}\)
Therefore,
\(\lim_{x\rightarrow\infty}\frac{3x^6-x^4+2x^2-3}{4x^5+2x^3-5}=\infty\)
0

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