tolove said:My book is showing this as an intuitive step, but I'm not quite seeing the reasoning behind it.
n**(c/n) → 1 as n → ∞, for, I think, any positive c. But why?
Dick said:Take the log and look at the limit of that. Use l'Hopital's rule, if it doesn't seem intuitive.
A limit with a square root is a mathematical concept that involves finding the value that a function approaches as its input (usually denoted as x) gets closer and closer to a specific value (usually denoted as a). This is represented as "lim f(x) as x approaches a" and is used to analyze the behavior of a function at a specific point.
A limit with a square root is evaluated by plugging in values of x that are close to the given value of a into the function. As the values of x get closer and closer to a, the resulting values of the function will also get closer and closer to the limit. If the function is undefined or has a non-real answer at a, the limit does not exist.
Yes, a limit with a square root can have a different value from the actual value of the function. This is because a limit only represents the behavior of a function at a specific point, and it does not necessarily reflect the overall behavior of the function. A function may have a discontinuity or a sharp turn at a specific point, but the limit at that point can still exist.
The common types of limits with square roots include square root limits, where the function has a square root in the numerator or denominator; rationalizing limits, where the function is multiplied or divided by its conjugate to eliminate square roots; and composition limits, where the function has a square root within a larger function.
Limits with square roots are important in mathematics because they help to understand the behavior of a function at a specific point, even if the function is undefined or has a non-real value at that point. They also help to determine the continuity and differentiability of a function, and are used in various applications in calculus and other areas of mathematics.