Question

1-2

1.Calculate the limit  

x + x2 - 5x+3 x-x-1 *

2.Given the limit: ,
a.Consider the boundary
b.A student claims that the limit is zero and justifies the following:
i. The first double strives endlessly
ii. The second double aims at zero
iii. Therefore the result is zero. Where exactly is the student wrong?

Answer 1

The first question can be solved by using L Hopital's rule for the evaluation of limits.

The second question can be solved by using the fact that the limits cannot be separated from their product form.

We have to find lim x²+x²–5x+3 XI x-x-1 - lim x²+x²–5x+3 xxl x- 1 Putting a find x=1, we get that the above limit is in the o form indeterminate type o. We may use Ľ Hopital's sule to evaluate : the required limit. Using L'Hopital's zule, we will differentiate both the numerator and the denominator separately and then check for the limit the putting n=l, until indeterminate form by vanishes. To lim = lim I (223+x²–5x+3) Using L'Hopital ule u annan - الا =lim 3x²+2x-5 *) = lim 3x²+ 20-5 211 it to Now, when we put nal, we find that it is NOT a form of indeterminate typer ilim atx²5n+3 - lim 3x²+2x-5 = 3 (1) 4 221-5 X- X1 it X

o The limit is o 2. (6) No, the student is wrong. lim adat lim ax lim & oo We cannot separate the limits . When they are in product forms xD bodo

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