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How do you find the limit of #((t^2) + 2)/(t-4)# as t approaches 4?

Answer 1

Limit does not exist

start by plugging in 4

#(4^2 + 2)/(4-4)# so we see that this goes to #oo# at #t = 4#

the numerator is always positive

....whereas the denominator is negative for #t = 4 - delta# and positive for #t = 4 + delta#, where #0 < delta " << " 1# so we have a two-sided limit

#lim_(t to 4^-) ((t^2) + 2)/(t-4) = -oo#

#lim_(t to 4^+) ((t^2) + 2)/(t-4) = oo#

Therefore limit does not exist

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Answer 2

Emma Aldridge

To find the limit of ((t^2) + 2)/(t-4) as t approaches 4, we can use direct substitution. However, direct substitution results in an undefined expression (division by zero). Therefore, we need to simplify the expression before evaluating the limit. By factoring the numerator, we get (t-2)(t+2)/(t-4). Now, we can cancel out the common factor of (t-4) in the numerator and denominator. This leaves us with (t+2) as the simplified expression. Finally, we can substitute t=4 into the simplified expression to find the limit. Thus, the limit of ((t^2) + 2)/(t-4) as t approaches 4 is 6.

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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.