Solving Limit as x approaches 3 using Multiplication and Division

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In summary, the conversation discusses solving the limit of a function involving square roots. One person suggests making the x - 3 in the numerator explicit and multiplying by $\sqrt{6x - 14} + \sqrt{x + 1}$, leading to a solution.
  • #1
tmt1
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I have to solve this limit.

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

Now, I think that by definition x - 3 is a divisor of the numerator, but how do I advance from here? Do I do long division?
 
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  • #2
tmt said:
I have to solve this limit.

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

Now, I think that by definition x - 3 is a divisor of the numerator, but how do I advance from here? Do I do long division?

How about making the x - 3 in the numerator explicit?
What do you get if you multiply both numerator and denominator by $\sqrt{6x - 14} + \sqrt{x + 1}$?
 
  • #3
I like Serena said:
How about making the x - 3 in the numerator explicit?
What do you get if you multiply both numerator and denominator by $\sqrt{6x - 14} + \sqrt{x + 1}$?

Okay thanks, now I got it.
 

FAQ: Solving Limit as x approaches 3 using Multiplication and Division

What does "Limit as x approaches 3" mean?

The limit as x approaches 3 is a mathematical concept that represents the value that a function approaches as its input variable, in this case x, gets closer and closer to the number 3.

How is the limit as x approaches 3 calculated?

The limit as x approaches 3 can be calculated by evaluating the function at values of x that are very close to 3, such as 2.9, 2.99, 2.999, and so on. If the function approaches the same value as x gets closer to 3, then that value is the limit.

What is the difference between a one-sided limit and a two-sided limit?

A one-sided limit only considers the values of the function as x approaches from one side, either the left or the right. A two-sided limit takes into account the values of the function as x approaches from both the left and the right.

Can the limit as x approaches 3 be undefined?

Yes, the limit as x approaches 3 can be undefined if the function has a discontinuity or a vertical asymptote at x=3, meaning that the function does not approach a specific value as x gets closer to 3.

Why is the limit as x approaches 3 important?

The limit as x approaches 3 is important because it helps us understand the behavior of a function near a specific point. It allows us to make predictions about the function and its values, even if the function is not defined at that point.

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