Evaluate numerator and denominator limit, Mathematics

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Evaluate following limits.

343_limit6.png

Solution : Let's do the first limit & in this case it sees like we will factor a z3 out of the numerator and denominator both. Remember that we just see at the denominator while determining the largest power of z here. In the numerator there is a larger power of z however we ignore it. We just see at the denominator while doing this! So doing the factoring gives,

1415_limit7.png

While we take the limit we'll have to be a little careful. The first term in the numerator & denominator will both be zero.  Though, the z3 in the numerator will be going to plus infinity in the limit and therefore the limit is,

1775_limit8.png

The final limit is negative since we have a quotient of positive quantity & a negative quantity.

Now, let's take a look at the second limit.  Note as well that the only different in the work is at the final "evaluation" step and therefore we'll pick up the work there.

1760_limit9.png

In this case the z3 in the numerator gives -ve infinity in the limit as we are going to minus infinity & the power is odd. The answer is positive as we have a quotient of two negative numbers.


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