p.ella said:Homework Statement
Evaluate the limit of each indeterminate quotient:
lim (x-->4) [2-(x^1/2)]/[3-(2x+1)^1/2]
Homework Equations
The Attempt at a Solution
The answer in the book is 3/4. This MAY be wrong though.
My attempt: I basically tried rationalizing the numerator AND denominator but got stuck here:
lim (x-->4) [12-3x+(4-x(2x+1)^1/2) ] / [16-4x+(8-2x(x)^1/2)]
Dick said:Next factor 4-x out of the numerator and denominator. And be careful. You are missing some parentheses there.
A limit is a mathematical concept that describes the behavior of a function as the input approaches a certain value. It is denoted by the symbol lim and is used to determine the value that a function approaches as the input gets closer and closer to a specific value.
To evaluate limits with radicals, you need to first simplify the expression by rationalizing the denominator and then substitute the given value in place of the variable. If the expression is still indeterminate, you can use algebraic manipulation or L'Hopital's rule to solve the limit.
The common indeterminate forms when evaluating limits with radicals are 0/0 and ∞/∞. These forms indicate that the limit cannot be determined without further manipulation or using additional techniques.
No, it is not possible to evaluate limits with radicals at points of discontinuity. This is because the function is not defined at these points, and the limit does not exist. To evaluate the limit at a point of discontinuity, you will need to use one-sided limits.
Graphs can help visualize the behavior of a function and provide an intuitive understanding of limits with radicals. By looking at the graph, you can see if the function approaches a specific value as the input gets closer to a certain value. You can also identify any points of discontinuity or vertical asymptotes that may affect the limit.