Help with continuity! "g(x)={sqrt(x^2+16) + x - 4 / x when x ≠ 0}..."

[Dhartju]

consider the function g(x)={sqrt(x^2+16) + x - 4 / x when x ≠ 0}
{ c when x = 0}
Determine what value of c, g(x) will be continuous at x = 0. Show your work.

okay so first, I tried to simplify the function by doing the limit as x approaches 0, and multiplying the top and bottom by sqrt(x^2+16) - x + 4.
Which then I get, (x^2+16) - x - 16 / x sqrt(x^2+16) - x + 4. By simplifying even more, I get = (x-1) / sqrt(x^2+16) - x + 4.
And now I am stuck. What do I do next??? Thank you in advance!!

 

[mmm4444bot]

consider the function g(x)={(sqrt(x^2+16) + x - 4) / x when x ≠ 0}
{ c when x = 0}
Determine what value of c, g(x) will be continuous at x = 0. Show your work.

When you type algebraic ratios, it's very important to use grouping symbols around numerators and denominators when they contain more than one term.

Without the grouping symbols shown in red above, your typing means this:

\(\displaystyle \sqrt{x^2+16} + x - \dfrac{4}{x}\)

okay so first, I tried to simplify the function by doing the limit as x approaches 0, and multiplying the top and bottom by sqrt(x^2+16) - x + 4. Good idea!

Which then I get, ((x^2+16) - x - 16) / ([x][sqrt(x^2+16) - x + 4]).

The numerator is not correct. Check to be sure that you completed all nine multiplications on top. The distribution takes this form:

(A + B + C)*(D + E + F) = AD + AE + AF + BD + BE + BF + CD + CE + CF

Once you correctly multiply and simplify, you will get n*x on top, where n is an Integer.

Do not multiply out the bottom because you want to cancel the factor of x on top with the factor of x on the bottom.

(I also inserted the missing brackets above which show the multiplication on the bottom.)

Now you can evaluate the limit by substitution. :cool:

 

[Dhartju]

When you type algebraic ratios, it's very important to use grouping symbols around numerators and denominators when they contain more than one term.

Without the grouping symbols shown in red above, your typing means this:

\(\displaystyle \sqrt{x^2+16} + x - \dfrac{4}{x}\)


The numerator is not correct. Check to be sure that you completed all nine multiplications on top. The distribution takes this form:

(A + B + C)*(D + E + F) = AD + AE + AF + BD + BE + BF + CD + CE + CF

Once you correctly multiply and simplify, you will get n*x on top, where n is an Integer.

Do not multiply out the bottom because you want to cancel the factor of x on top with the factor of x on the bottom.

(I also inserted the missing brackets above which show the multiplication on the bottom.)

Now you can evaluate the limit by substitution. :cool:

Hello, Thank you for your reply and help! Finally see what I did wrong! Thank you!