What is the limit as x approaches infinity of (5000)/(20+30e^{-0.03x)} ?

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limit as x approaches infinity of (5000)/(20+30e^{-0.03x)}

x \to \infty lim (\frac{5000}{20 + 30 e ^{- 0.03 x}})

250

Solution steps

x \to \infty lim (\frac{5000}{20 + 30 e ^{- 0.03 x}})

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

= 5000 \cdot x \to \infty lim (\frac{1}{20 + 30 e ^{- 0.03 x}})

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

= 5000 \cdot \frac{lim _{x \to \infty} ( 1 )}{lim _{x \to \infty} ( 20 + 30 e ^{- 0.03 x} )}

x \to \infty lim (1) = 1

x \to \infty lim (20 + 30 e^{- 0.03 x}) = 20

= 5000 \cdot \frac{1}{20}

Simplify

5000 \cdot \frac{1}{20} : 250

= 250

What is the limit as x approaches infinity of (5000)/(20+30e^{-0.03x)} ?
The limit as x approaches infinity of (5000)/(20+30e^{-0.03x)} is 250