# For what values of x, if any, does #f(x) = 1/((x-1)(x-7)) # have vertical asymptotes?

x = 1 , x = 7

The denominator of the rational function cannot equal zero as this would lead to division by zero which is undefined. Setting the denominator equal to zero and solving for x will give the values that x cannot be and if the numerator is non-zero for these values of x then they must be vertical asymptotes.

solve: (x-1)(x-7) = 0 → x = 1 , x = 7

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The function f(x) = 1/((x-1)(x-7)) has vertical asymptotes at x = 1 and x = 7.

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The function ( f(x) = \frac{1}{(x-1)(x-7)} ) will have vertical asymptotes where the denominator, ( (x-1)(x-7) ), equals zero.

Setting each factor equal to zero and solving for ( x ) gives us the values where vertical asymptotes occur:

- ( x - 1 = 0 ) gives ( x = 1 ).
- ( x - 7 = 0 ) gives ( x = 7 ).

Thus, the function has vertical asymptotes at ( x = 1 ) and ( x = 7 ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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