The 2 functions ( (x^2 – 4)/(x-2) vs. x+2 ) aren’t equivalent since one is undefined at x=2 and the other isn’t, but you can say their limits are equal since disagreeing at a single point doesn’t change the limit.
Daniel Muñoz says:
“Since disagreeing at a single point doesn’t change the limit” <- That's not exact. Since the rational function is not continuous on x = 2, by definition of limit we don't care which is the value of f in x = 2. It's not just about disagreeing at a point.
Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment): Cancel reply
![\[ \lim_{x \to 2} \frac{x^2 - 4}{x-2}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-43572b335565f01d1f92f424df6fe28f_l3.png "Rendered by QuickLaTeX.com")
![\[ \frac{x^2-4}{x-2} = \frac{(x+2)(x-2)}{x-2} = x+2. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-8c379bedeab9afb18cbaab08c741425f_l3.png "Rendered by QuickLaTeX.com")
![\[ \lim_{x \to 2} (x+2) = 4. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-15da5277e880af70d9b3b1f08a84d385_l3.png "Rendered by QuickLaTeX.com")
The 2 functions ( (x^2 – 4)/(x-2) vs. x+2 ) aren’t equivalent since one is undefined at x=2 and the other isn’t, but you can say their limits are equal since disagreeing at a single point doesn’t change the limit.
“Since disagreeing at a single point doesn’t change the limit” <- That's not exact. Since the rational function is not continuous on x = 2, by definition of limit we don't care which is the value of f in x = 2. It's not just about disagreeing at a point.