What is the limit as x approaches infinity of ((ax+b)^2)/((a-x)(b-x)) ?

The limit as x approaches infinity of ((ax+b)^2)/((a-x)(b-x)) is a^2

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limit as x approaches infinity of ((ax+b)^2)/((a-x)(b-x))

x \to \infty lim (\frac{( a x + b ) ^{2}}{( a - x ) ( b - x )})

a^{2}

Solution steps

x \to \infty lim (\frac{( a x + b ) ^{2}}{( a - x ) ( b - x )})

Apply the following algebraic property

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

\frac{( a x + b ) ^{2}}{( a - x ) ( b - x )} = \frac{( x ( a + \frac{b}{x} ) ) ^{2}}{( x ( \frac{a}{x} - 1 ) ) ( x ( \frac{b}{x} - 1 ) )}

= x \to \infty lim (\frac{( x ( a + \frac{b}{x} ) ) ^{2}}{x ( \frac{a}{x} - 1 ) x ( \frac{b}{x} - 1 )})

Simplify

\frac{( x ( a + \frac{b}{x} ) ) ^{2}}{( x ( \frac{a}{x} - 1 ) ) ( x ( \frac{b}{x} - 1 ) )} : \frac{( a + \frac{b}{x} ) ^{2}}{( \frac{a}{x} - 1 ) ( \frac{b}{x} - 1 )}

= x \to \infty lim (\frac{( a + \frac{b}{x} ) ^{2}}{( \frac{a}{x} - 1 ) ( \frac{b}{x} - 1 )})

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 _{x \to \infty} ( ( a + \frac{b}{x} ) ^{2} )}{lim _{x \to \infty} ( ( \frac{a}{x} - 1 ) ( \frac{b}{x} - 1 ) )}

x \to \infty lim ((a + \frac{b}{x})^{2}) = a^{2}

x \to \infty lim ((\frac{a}{x} - 1) (\frac{b}{x} - 1)) = 1

= \frac{a ^{2}}{1}

Simplify

= a^{2}

x \to 0 lim (3 tan (x) ln (x))

x \to 3 - lim (\frac{3 - x}{∣ x - 3 ∣})

x \to + 0 lim (\frac{tan ( 8 x )}{x})

x \to 8 lim (2 x - 8)

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

What is the limit as x approaches infinity of ((ax+b)^2)/((a-x)(b-x)) ?
The limit as x approaches infinity of ((ax+b)^2)/((a-x)(b-x)) is a^2