The limit of the function f(x) = x² - 10x + 25 / x-5 as x approaches 5 is undefined. This is because when x approaches 5, the denominator becomes 0, which is an undefined value in mathematics.
To evaluate the limit of the function f(x) = x² - 10x + 25 / x-5, you can use the concept of factorization. By factoring the numerator, you can simplify the function to f(x) = (x-5)(x-5) / x-5. Then, you can cancel out the common factors of x-5 to get f(x) = x-5. Finally, you can substitute the value of 5 for x to get the limit as 0.
No, the limit of the function f(x) = x² - 10x + 25 / x-5 cannot be evaluated using direct substitution. As mentioned earlier, this is because when x approaches 5, the denominator becomes 0, which is an undefined value. Direct substitution can only be used when the function is defined and continuous at the given value.
The function being undefined at x=5 means that the function is not continuous at this point. This can indicate a potential break or discontinuity in the graph of the function, which may affect the overall behavior of the function.
Yes, it is possible to manipulate the function to make the limit defined at x=5. In this case, you can use the concept of a removable discontinuity. By simplifying the function and canceling out the common factors of x-5, you can create a new function that is defined and continuous at x=5. However, this new function may have a different value compared to the original function at other points, so it is important to consider the overall behavior of the function before manipulating it.