How do you find #lim (5+x^-1)/(1+2x^-1)# as #x->oo# using l'Hospital's Rule or otherwise?

Answer 1

# lim_( x rarr oo) (5+x^-1)/(1+2x^-1) =5#

In its present form it should be clear that you do not need to apply L'Hôpital's rule as #x^-1 rarr 0# as #x rarr oo# and so;
# lim_( x rarr oo) (5+x^-1)/(1+2x^-1) = (5+0)/(1+0) =5#
If you do want to apply L'Hôpital's rule the we should multiply numerator and denominator both by #x# to get;
# lim_( x rarr oo) (5+x^-1)/(1+2x^-1) = lim_( x rarr oo) x/x(5+x^-1)/(1+2x^-1)# # " " = lim_( x rarr oo) (5x+1)/(x+2)#
We now have an indeterminate form of the type #oo/oo# so we can apply L'Hôpital's rule:
# lim_(x rarr a) f(x)/g(x) = lim_(x rarr a) (f'(x))/(g'(x)) #

having satisfied ourselves that our limit meets L'Hôpital's criteria to get

# lim_( x rarr oo) (5+x^-1)/(1+2x^-1) = lim_( x rarr oo) (d/dx(5x+1))/(d/dx(x+2))# # " " = lim_( x rarr oo) 5/1 = 5#, as above.
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Answer 2

To find ( \lim_{x \to \infty} \frac{5 + x^{-1}}{1 + 2x^{-1}} ) using l'Hôpital's Rule:

  1. Rewrite the expression as ( \frac{5x + 1}{x + 2} ) since ( x^{-1} = \frac{1}{x} ).

  2. Apply l'Hôpital's Rule by taking the derivatives of the numerator and denominator with respect to ( x ).

  3. Differentiate the numerator: ( (5x + 1)' = 5 ).

  4. Differentiate the denominator: ( (x + 2)' = 1 ).

  5. Evaluate the limit as ( x ) approaches infinity: ( \lim_{x \to \infty} \frac{5}{1} = 5 ).

Alternatively, you can directly evaluate the limit by observing that as ( x ) approaches infinity, the terms ( x^{-1} ) become negligible compared to the constant terms, resulting in ( \frac{5}{1} = 5 ).

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Answer from HIX Tutor

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.

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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