Finding limit of a funciton with square roots.

[tmt1]

I have to find this:

$$\lim_{{x}\to{3}}\frac {\sqrt{6x - 14} - \sqrt { x+1}}{x-3}$$

So I do this:

$$\lim_{{x}\to{3}}\frac {\sqrt{6x - 14} - \sqrt { x+1}}{x-3} * \frac{\sqrt{6x + 14} + \sqrt{x+1}}{\sqrt{6x + 14} + \sqrt{x+1}}$$

The top part is easy since

$$(\sqrt{a} - \sqrt{b})(\sqrt{a} + \sqrt{b}) $$ is equal to $$a - b$$

So I have

$$\frac {5x- 14}{x\sqrt{6x+4} + x\sqrt{x+1} - 3\sqrt{6x+4} - 3\sqrt{x+1}} $$

From here I am lost.

 

[Evgeny.Makarov]

So I have

$$\frac {5x- 14}{x\sqrt{6x+4} + x\sqrt{x+1} - 3\sqrt{6x+4} - 3\sqrt{x+1}} $$

Then you divide the numerator and denominator by $x$ and find their limits separately.

Edit: Sorry, I missed the fact that the limit is when $x\to3$; I thought it is when $x\to\infty$.

 

[Dethrone]

Consider:

$$=\lim_{{x}\to{3}}\frac {\sqrt{6x - 14} - \sqrt { x+1}}{x-3}$$
$$=\lim_{{x}\to{3}}\frac {\sqrt{6x - 14}-2 - \sqrt { x+1}+2}{x-3}$$
$$=\lim_{{x}\to{3}}\frac {\sqrt{6x - 14}-2 - (\sqrt { x+1}-2)}{x-3}$$
$$=\lim_{{x}\to{3}}\frac{\sqrt{6x-14}-2}{x-3}-\frac{\sqrt{x+1}-2}{x-3}$$
$$=\d{}{x}\sqrt{6x-14}|_{x=3}-\d{}{x}\sqrt{x+1}|_{x=3}$$

The rest is easy :D

 

[soroban]

Hello, tmt!

Find: $\displaystyle \lim_{x\to3}\frac {\sqrt{6x - 14} - \sqrt { x+1}}{x-3}$

$\displaystyle \lim_{{x}\to{3}}\frac {\sqrt{6x - 14} - \sqrt { x+1}}{x-3} * \frac{\sqrt{6x - 14} + \sqrt{x+1}}{\sqrt{6x - 14} + \sqrt{x+1}}$

$\displaystyle \quad=\; \lim_{x\to3}\frac{(6x-14) - (x+1)}{(x-3)(\sqrt{6x-14} + \sqrt{x+1})} \;=\; \lim_{x\to3}\frac{5x-15}{(x-3)(\sqrt{6x-14} + \sqrt{x+1}}$

$\displaystyle \quad=\;\lim_{x\to3}\frac {5(x- 3)}{(x-3)(\sqrt{6x-14} + \sqrt{x+1}} \;=\; \lim_{x\to3}\frac{5}{\sqrt{6x-14} + \sqrt{x+1}}$

Can you finish it now?

 

[CellCoree]

I would commend the individual for their approach in solving the limit of the given function with square roots. It is clear that they have a good understanding of the properties of square roots and how to manipulate them in order to simplify the expression.

However, I would also suggest that they continue working on the problem to see if they can simplify the expression further. It may be helpful to try factoring out common terms or using algebraic techniques such as conjugate multiplication to eliminate the square roots in the denominator.

In addition, I would advise the individual to consider the properties of limits and how they can be used to evaluate the given function. For example, they could use the fact that the limit of a sum is equal to the sum of the limits, or that the limit of a product is equal to the product of the limits.

Overall, the individual has made a good start in solving the limit of the given function with square roots, but I would encourage them to continue exploring different techniques and properties in order to find a simplified expression and ultimately determine the limit.