Find the limit of the function algebraically (with cancellation techniques and direct substitution, that is, since this is in indeterminate form): it seems to have a dividing by zero issue from what i have managed to get out of the that or i am having trouble discerning how to break it down into a manageable 0 issue. would be greatly appreciated, or at the very least a confirmation that i have the correct answer in that there is no limit.

At x=0, the numerator is positive and the denominator is zero. The form is not "indeterminate", but is "undefined". The limit is the same (+∞) whether approaching zero from the left or the right, so we say

  the limit is +∞


The case of "no limit" is reserved for those conditions in which the value you get approching from the left is different from that approaching from the right and/or is different from the defined value of the function at the point.

For example, 1/x has no limit at x=0 because it approaches -∞ from the left and +∞ from the right.

The null and alternative hypotheses:

H0: attitudes between first survey and second survey have not changed.

H1:attitudes between first survey and second survey have changed.

Let's find the observed and expected value using the following:

Observed value = sample size * observed proportion

Expected value = sample size * expected proportion.

Optimistic:

expected= 6%*380=22.8

observed=1%*380=3.8  

(fo-fe)²/fe = 15.83333

Slightly Optimistic:

expected: 29%*380=110.2);

Observed: 30%*380= 114

(fo-fe)²/fe  = 0.131034

Slightly Pessimistic:

expected= 24%*380= 91.2

Observed = 28%*380=106.4  

(fo-fe)²/fe =  2.533333

Pessimistic:

Expected = 41%*380 =155.8

Observed = 41%*380 =155.8

(fo-fe)²/fe = 0

Total (fo-fe)²/fe =

15.83333 + 0.131034 + 2.533333 + 0 = 18.4977

Degrees of freedom, df = 4 - 1 = 3

For chi square test statistic:

Σ(fo-fe)²/fe = 15.83333 + 0.131034 + 2.533333 + 0 =  18.498

Calculate pvalue.

Calculating for pvalue at, df = 3, chi-square score = 18.4977, α=0.05

Pvalue = 0.000347

Decision:

Since pvalue, 0.000347 is less than significance level, 0.05, reject null hypothesis H0.

Conclusion:

There is enough evidence to conclude that there is a change in attitudes between  first survey and the second survey at a level of significance, 0.05.