What is the limit as z approaches infinity of (4z^5)/(z^5-42z) ?

The limit as z approaches infinity of (4z^5)/(z^5-42z) is 4

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limit as z approaches infinity of (4z^5)/(z^5-42z)

z \to \infty lim (\frac{4 z ^{5}}{z ^{5} - 42 z})

4

Solution steps

z \to \infty lim (\frac{4 z ^{5}}{z ^{5} - 42 z})

x \to a lim [c \cdot f (x)] = c \cdot x \to a lim f (x)

= 4 \cdot z \to \infty lim (\frac{z ^{5}}{z ^{5} - 42 z})

Simplify

\frac{z ^{5}}{z ^{5} - 42 z} : \frac{z ^{4}}{z ^{4} - 42}

= 4 \cdot z \to \infty lim (\frac{z ^{4}}{z ^{4} - 42})

Divide by highest denominator power:

\frac{1}{1 - \frac{42}{z ^{4}}}

= 4 \cdot z \to \infty lim (\frac{1}{1 - \frac{42}{z ^{4}}})

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

= 4 \cdot \frac{lim _{z \to \infty} ( 1 )}{lim _{z \to \infty} ( 1 - \frac{42}{z ^{4}} )}

z \to \infty lim (1) = 1

z \to \infty lim (1 - \frac{42}{z ^{4}}) = 1

= 4 \cdot \frac{1}{1}

Simplify

= 4

What is the limit as z approaches infinity of (4z^5)/(z^5-42z) ?
The limit as z approaches infinity of (4z^5)/(z^5-42z) is 4