Question about Direct Substitution when Computing Limits

[Laucchi]

Find the limit of the square root of (x^2+3x-4)/2 as x approaches 2.
I used direct substitution, and my answer was the square root of 3. Is it appropriate to use direct substitution in this case, or do I need to use something else?
Thank you!

 

[HallsofIvy]

As long as a function is continuous, yes, \(\displaystyle \lim_{x\to a} f(x)= f(a)\). In fact, that is the definition of continuous. And this function is continuous wherever it is defined, which is all \(\displaystyle x\ge 1\) or all \(\displaystyle x\le -4\). Since that includes s= 2, the limit is just its value at x= 2.


(All polynomials are continuous for all x, rational functions are continuous as long as the denominator is not 0, and roots are continuous wherever they are defined.)

 

[lookagain]

As long as a function is continuous, yes, \(\displaystyle \lim_{x\to a} f(x)= f(a)\).
In fact, that is the definition of continuous. And this function is continuous wherever it is defined,
which is all \(\displaystyle x\ge 1\) or all \(\displaystyle x\le -4\). Since that includes s= 2, the limit is just its
value at x= 2.


(All polynomials are continuous for all x, rational functions are continuous as long as the denominator
is not 0, and roots are continuous wherever they are defined.)

HallsofIvy,

this is a reminder that the Latex won't show up in this particular mathematics forum with the
itex, /itex formatting.

 

[mmm4444bot]

I think Halls understands the itex issue; he needs reminding to use the [Preview Post] button, instead.

I suffer from similar brain infarcts -- eg: some of my math sofware inserts the ( when I select sqrt() function, where other software inserts only sqrt and I must type both parentheses. I'm always getting syntax errors because I cannot seem to break old habits when switching from one software to another!

 

[HallsofIvy]

Mea Culpa,
Mea Culpa,
Mea Maxima Culpa.