For the function f(x) = #sqrtx+1/sqrtx# what are the intercepts and asymptotes ?
It does not intercept the
Given:
Note that:
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The function ( f(x) = \frac{\sqrt{x} + 1}{\sqrt{x}} ) has no intercepts.
As for asymptotes, there is a vertical asymptote at ( x = 0 ) because the denominator approaches zero as ( x ) approaches zero. There are no horizontal or slant asymptotes.
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To find the intercepts and asymptotes of the function ( f(x) = \frac{\sqrt{x+1}}{\sqrt{x}} ), we need to analyze its behavior.

Intercepts:
 yintercept: To find the yintercept, set ( x = 0 ) and evaluate ( f(x) ). ( f(0) = \frac{\sqrt{0+1}}{\sqrt{0}} = \frac{\sqrt{1}}{0} = 1 ) So, the yintercept is at (0, 1).
 xintercept: To find the xintercept, set ( f(x) = 0 ) and solve for ( x ). ( 0 = \frac{\sqrt{x+1}}{\sqrt{x}} ) This equation has no real solutions, so there are no xintercepts.

Asymptotes:
 Vertical asymptote: Vertical asymptotes occur where the function approaches infinity or negative infinity as ( x ) approaches a certain value. In this case, the function has a vertical asymptote at ( x = 0 ) because the denominator becomes zero and the function approaches infinity as ( x ) approaches 0 from the right.
 Horizontal asymptote: To find horizontal asymptotes, we analyze the behavior of the function as ( x ) approaches positive or negative infinity. ( \lim_{{x \to \infty}} f(x) = \lim_{{x \to \infty}} \frac{\sqrt{x+1}}{\sqrt{x}} = \lim_{{x \to \infty}} \frac{\sqrt{x} \sqrt{1+\frac{1}{x}}}{\sqrt{x}} = \lim_{{x \to \infty}} \sqrt{1 + \frac{1}{x}} = \sqrt{1 + 0} = 1 ) ( \lim_{{x \to \infty}} f(x) = \lim_{{x \to \infty}} \frac{\sqrt{x+1}}{\sqrt{x}} = \lim_{{x \to \infty}} \frac{\sqrt{x} \sqrt{1+\frac{1}{x}}}{\sqrt{x}} = \lim_{{x \to \infty}} \sqrt{1 + \frac{1}{x}} = \sqrt{1 + 0} = 1 ) Therefore, the function has a horizontal asymptote at ( y = 1 ).
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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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