Evaluate:
![\[ \lim_{x \to 2} \frac{1}{x^2}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-b77400c88e831b9085babe146dfccebd_l3.png "Rendered by QuickLaTeX.com")
We know that constant functions and polynomials are continuous (from the examples in Apostol sections 3.3 and 3.4). So, we have
![\[ \lim_{x \to 2} 1 = 1, \qquad \lim_{x \to 2} x^2 = 4. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-e8c92aded9b90f8adc029962d6488ea0_l3.png "Rendered by QuickLaTeX.com")
We may then apply Theorem 3.2 to conclude the quotient 
is continuous as long as 
. Thus,
![\[ \lim_{x \to 2} \frac{1}{x^2} = \frac{1}{2^2} = \frac{1}{4}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-aec09959516282a8014801203656b8a6_l3.png "Rendered by QuickLaTeX.com")
Evaluate:
We know that constant functions and polynomials are continuous (from the examples in Apostol sections 3.3 and 3.4). So, we have
We may then apply Theorem 3.2 to conclude the quotient is continuous as long as . Thus,