Finding limits of multiple functions

[Sam007]

Hi everyone,

I just wanted to confirm that the answers to the question is

a. 4
b. 1
c. -1/4
d. -37/13


If I am wrong, your advice would be greatly appreciated

Also for finding limits in general. Is it ok to sub the value x is approaching to find the limit

 

[Otis]

… just wanted to confirm … answers …

…
c. -1/4
d. -37/13
…

Hello. In your other thread, you said the double-sided limit as x→3 existed because the function approaches the same y-value, as x approaches 3 from both the left and the right. Now you're saying the two y-values (above) are different. Check the graph again; the limit is the same when x→3^- as it is when x→3⁺.

Perhaps you forgot the Order of Operations, when evaluating the expression: -37/4 + x^2

Do division before addition. In other words, read that expression as: -\(\frac{37}{4} + x^{2}\).

Regarding your question about whether we can generally find limits by directly substituting the boundary-value for x, the answer is no. In fact, the opposite is true. In general, we cannot find limits by direct substitution.

?

 

[JeffM]

For many functions, you can find what the limit is at most (or even all) values in the domain simply by substitution, BUT

(1) This NEVER proves that what you found is the limit, and

(2) It is the exceptions that are usually important.

With respect to the latter point, consider

[MATH]\lim_{x \rightarrow 0} \dfrac{sin(x)}{x}.[/MATH]
How does substitution of 0 possibly help you? It gives you

[MATH]\dfrac{0}{0}.[/MATH]
Virtually all of calculus is based on such exceptions.