iloveannaw said:Homework Statement
the title says it all
[tex]x \rightarrow 4[/tex]
for [tex]f\left(x\right) = \frac{\sqrt{1+2x}-3}{\sqrt{x}-2}[/tex]
I have multiplied both top and bottom by conjugate, [tex]\sqrt{x}+2[/tex]:
[tex]f\left(x\right) = \frac{\sqrt{x(1+2x)}+2\sqrt{1+2x} -3\sqrt(x)-6}{x-4}[/tex]
but don't know how to take this further. Dividing both numerator and denominator by x doesn't help.
A limit of a function with rational is the value that a function approaches as the input approaches a certain value, also known as the limit point. This value can be determined by evaluating the function at values that are closer and closer to the limit point.
The limit of a function with rational can be calculated by using the algebraic definition of a limit, which involves evaluating the function at values that are closer and closer to the limit point, and then simplifying the resulting expression.
Limits of functions with rational are closely related to continuity. A function is said to be continuous at a point if the limit of the function at that point exists and is equal to the value of the function at that point. If a function is not continuous at a point, it means that the limit of the function at that point does not exist or is not equal to the value of the function at that point.
Yes, a limit of a function with rational can be undefined. This can happen if the function has a vertical asymptote at the limit point, meaning that the function approaches positive or negative infinity as the input approaches the limit point.
Limits of functions with rational have many applications in the real world. For example, they can be used in engineering to determine the maximum load a structure can withstand, in physics to calculate instantaneous velocity and acceleration, and in economics to analyze supply and demand functions. They are also used in many other fields to model and analyze real-world phenomena.