What is the limit as z approaches infinity of ((az+b)^3)/((cz+d)^3) ?

The limit as z approaches infinity of ((az+b)^3)/((cz+d)^3) is (a^3)/(c^3)

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limit as z approaches infinity of ((az+b)^3)/((cz+d)^3)

z \to \infty lim (\frac{( a z + b ) ^{3}}{( cz + d ) ^{3}})

\frac{a ^{3}}{c ^{3}}

Solution steps

z \to \infty lim (\frac{( a z + b ) ^{3}}{( cz + d ) ^{3}})

Apply the following algebraic property

: a + b = a (1 + \frac{b}{a})

\frac{( a z + b ) ^{3}}{( cz + d ) ^{3}} = \frac{( z ( a + \frac{b}{z} ) ) ^{3}}{( z ( c + \frac{d}{z} ) ) ^{3}}

= z \to \infty lim (\frac{( z ( a + \frac{b}{z} ) ) ^{3}}{( z ( c + \frac{d}{z} ) ) ^{3}})

Simplify

\frac{( z ( a + \frac{b}{z} ) ) ^{3}}{( z ( c + \frac{d}{z} ) ) ^{3}} : \frac{( a + \frac{b}{z} ) ^{3}}{( c + \frac{d}{z} ) ^{3}}

= z \to \infty lim (\frac{( a + \frac{b}{z} ) ^{3}}{( c + \frac{d}{z} ) ^{3}})

x \to a lim [\frac{f ( x )}{g ( x )}] = \frac{lim _{x \to a} f ( x )}{lim _{x \to a} g ( x )}, x \to a lim g (x) \neq = 0

With the exception of indeterminate form

= \frac{lim _{z \to \infty} ( ( a + \frac{b}{z} ) ^{3} )}{lim _{z \to \infty} ( ( c + \frac{d}{z} ) ^{3} )}

z \to \infty lim ((a + \frac{b}{z})^{3}) = a^{3}

z \to \infty lim ((c + \frac{d}{z})^{3}) = c^{3}

= \frac{a ^{3}}{c ^{3}}

x \to 3 lim (4^{x})

x \to 0 lim (ln (x) cos (x))

x \to \infty lim (- x^{- 1})

x \to 0 lim (\frac{- e ^{x}}{x ^{2}})

x \to - 1 + lim (\frac{∣ x + 1 ∣}{x + 1})

What is the limit as z approaches infinity of ((az+b)^3)/((cz+d)^3) ?
The limit as z approaches infinity of ((az+b)^3)/((cz+d)^3) is (a^3)/(c^3)