How do you solve the following limit without a calculator?

[KataKoniK]

Hi,

I was wondering, how would one solve the following equation without using a calculator. In other words, algebraically.

lim (x + sqrt(x^2+5x))
x-> -infinity

Thanks in advance

 

[arildno]

Multiply with the conjugate expression:
$$x+\sqrt{x^{2}+5x}=(x+\sqrt{x^{2}+5x})\frac{x-\sqrt{x^{2}+5x}}{x-\sqrt{x^{2}+5x}}=-\frac{5x}{x-\sqrt{x^{2}+5x}}\to-\frac{5}{2}, x\to\infty$$

 

[KataKoniK]

Thanks a lot. Really appreciate it. I thought you had to do it a certain way because the limit is approaching infinity instead of a number.

 

[KataKoniK]

Didn't notice this, but how does the bottom become 2?

 

[marlon]

Didn't notice this, but how does the bottom become 2?

Because when calculating the limit to -infinity you need to put the denominator $$x-\sqrt{x^{2}+5x}$$ in factorized form. When doing so you need to get an x² out of the square-root but realize that x is negative so you need to write $$x-(-x)\sqrt{1+\frac{5x}{x^2}}$$. This is just like saying that $$\sqrt{9} = \sqrt{(-3)(-3)} = -3$$. Factoring on you will get that $$x(1+\sqrt{1+\frac{5}{x}})$$ and the x will vanish because of the x you will get in the nominator after completing the exact same procedure there. If you fill in $$- \infty$$ you will get the 2 in the bottom


regards
marlon

 

[KataKoniK]

Thank you!