Find VA and HA for f(x) = (a^-1 + x^-1)^-1 where a is a positive number.
I have to find it algebraically.
I have to find it algebraically.
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to find the horizontal tangent, as the following:
f (x) = (a^-1 + x^-1)^-1
f (x) = ( (1 / a) + (1 / x) )^-1
f (x) = [ 1 ] / [ ( (1*x / a*x) + (1*a / x*a) ) ]
f (x) = [ 1 ] / [ ( (x / a*x) + (a / x*a) ) ]
f (x) = [ 1 ] / [ ( x + a )/ (a*x) ]
f (x) = (a * x) / ( x + a ) ====> since the numerator and denominator have the same degree, we can find it as the following:
f (x) = (a * x) / ( x ) = a ( horizontal tangent at a )
TO DETERMINE THE VERTICAL ASYMPTOTE, we will check the denominator of the following:
f (x) = (a * x) / ( x + a ) ====> let's only equal the denominator to zero as the following:
x + a = 0 ====> x = -a .....so only x can't be -a
domain of the vertical asymptote as (-∞ , -a) U (-a , ∞)
f (x) = (a^-1 + x^-1)^-1
f (x) = ( (1 / a) + (1 / x) )^-1
f (x) = [ 1 ] / [ ( (1*x / a*x) + (1*a / x*a) ) ]
f (x) = [ 1 ] / [ ( (x / a*x) + (a / x*a) ) ]
f (x) = [ 1 ] / [ ( x + a )/ (a*x) ]
f (x) = (a * x) / ( x + a ) ====> since the numerator and denominator have the same degree, we can find it as the following:
f (x) = (a * x) / ( x ) = a ( horizontal tangent at a )
TO DETERMINE THE VERTICAL ASYMPTOTE, we will check the denominator of the following:
f (x) = (a * x) / ( x + a ) ====> let's only equal the denominator to zero as the following:
x + a = 0 ====> x = -a .....so only x can't be -a
domain of the vertical asymptote as (-∞ , -a) U (-a , ∞)