Let #h(t) = (2t)/(t)# and #g(t)= (3t+15)/t#, how do you find (h/g)(t) and the domain for h/g?
We have that
We have that
The domain of a quotient is a subset of the intersection of the domains. Since neither domain contains 0, the domain of the quotient excludes 0 as well.
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To find ( \frac{h(t)}{g(t)} ) and determine its domain, follow these steps:

Write the Expression for ( \frac{h(t)}{g(t)} ): [ \frac{h(t)}{g(t)} = \frac{\frac{2t}{t}}{\frac{3t+15}{t}} ]

Simplify the Expression: [ \frac{h(t)}{g(t)} = \frac{2t}{3t+15} ]

Determine the Domain: The domain is determined by the values of ( t ) for which the expression is defined. In this case, since we have a rational expression, we need to exclude values of ( t ) that make the denominator zero. [ 3t + 15 = 0 ] [ 3t = 15 ] [ t = 5 ]
Thus, the domain for ( \frac{h(t)}{g(t)} ) is all real numbers except ( t = 5 ), or in interval notation: ( (\infty, 5) \cup (5, \infty) ).
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To find ((h/g)(t)), we divide (h(t)) by (g(t)) termwise.
[ \frac{h(t)}{g(t)} = \frac{\frac{2t}{t}}{\frac{3t+15}{t}} = \frac{2t}{3t+15} ]
The domain for ((h/g)(t)) will be the intersection of the domains of (h(t)) and (g(t)), excluding any values of (t) that make the denominator of (g(t)) equal to zero. Since the denominator of (g(t)) is (t), we cannot have (t = 0). Therefore, the domain for ((h/g)(t)) is all real numbers except (t = 0).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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