How do you find the asymptotes for #y=x*ln[e + (1/x)] #?
Asymptote at
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To find the asymptotes of the function ( y = x \cdot \ln\left(e + \frac{1}{x}\right) ), we need to examine the behavior of the function as ( x ) approaches certain values.

Vertical Asymptote: As ( x ) approaches zero from the positive side (( x \rightarrow 0^+ )), the term ( \frac{1}{x} ) approaches positive infinity, and consequently, ( \ln(e + \frac{1}{x}) ) approaches positive infinity. Thus, there is a vertical asymptote at ( x = 0 ).

Horizontal Asymptote: As ( x ) approaches positive or negative infinity (( x \rightarrow \pm \infty )), the term ( \frac{1}{x} ) approaches zero, and ( \ln(e + \frac{1}{x}) ) approaches ( \ln(e) = 1 ). Therefore, there is a horizontal asymptote at ( y = x \cdot 1 = x ).
Thus, the asymptotes for the function ( y = x \cdot \ln\left(e + \frac{1}{x}\right) ) are:
 Vertical asymptote: ( x = 0 )
 Horizontal asymptote: ( y = x )
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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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