How do you find the limit of #(x^2+3)/(x^2+4)# as #x->-oo#?

Answer 1

# lim_(x->-oo)(x^2+3)/(x^2+4) = 1 #

If we look at the graph of #y=(x^2+3)/(x^2+4)# we can see that it is clear that the limit exists, and is approximately #1#

graph{[-10, 10, -2, 2]} graph{(x^2+3)/(x^2+4)

Now, As #x->-oo# then #x^2->oo# ,but #1/x^2->0#
So, it would be better if we could replace #x^2# with #1/x^2#, or #x^-2#
# lim_(x->-oo)(x^2+3)/(x^2+4) = lim_(x->-oo)((x^2+3))/((x^2+4)) * x^-2/x^-2 #
# :. lim_(x->-oo)(x^2+3)/(x^2+4) = lim_(x->-oo)(x^-2(x^2+3))/(x^-2(x^2+4)) #
# :. lim_(x->-oo)(x^2+3)/(x^2+4) = lim_(x->-oo) (x^-2x^2+3x^-2) / (x^-2x^2+4x^-2)#
# :. lim_(x->-oo)(x^2+3)/(x^2+4) = lim_(x->-oo) (1+3x^-2) / (1+4x^-2)#
And, using #x^-2->0# as #x->-oo# we have;
# lim_(x->-oo)(x^2+3)/(x^2+4) = (1+0) / (1+0) = 1#

which agrees with the graph above perfectly.

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Answer 2

# lim_(x->-oo)(x^2+3)/(x^2+4) = 1 #

We can also use L'Hospital's Rule as the limit is of an indeterminate form #oo/oo#

In light of L'Hospital's Rule,

# lim_(x->-oo)(x^2+3)/(x^2+4) = lim_(x->-oo)((x^2+3)')/((x^2+4)') # # :. lim_(x->-oo)(x^2+3)/(x^2+4) = lim_(x->-oo)(2x)/(2x) # # :. lim_(x->-oo)(x^2+3)/(x^2+4) = lim_(x->-oo)(1) # # :. lim_(x->-oo)(x^2+3)/(x^2+4) = 1 #
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Answer 3

To find the limit of (x^2+3)/(x^2+4) as x approaches negative infinity, we can divide both the numerator and denominator by x^2. This gives us the limit of (1 + 3/x^2)/(1 + 4/x^2) as x approaches negative infinity. As x approaches negative infinity, both 3/x^2 and 4/x^2 approach zero. Therefore, the limit simplifies to 1/1, which is equal to 1. Hence, the limit of (x^2+3)/(x^2+4) as x approaches negative infinity is 1.

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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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