Question

Evaluate algebraically?

Answer 1

By eliminating the evitable discontinuity, the limit of the rational function when x tends to 4 is equal to 0.

How to determine the limit of a rational function with radical elements with evitable discontinuity

In this problem we find the case of a rational function with radical elements whose limit apparently does not exist. Nonetheless, we can also try to eliminate the discontinuity by factorization:

First, write the entire expression:

`\lim_(x \to 4) (x - 4)/(√(x) - √(8 - x))`

Second, use factorization:

`\lim_(x \to 4) ((x - 4)\cdot (√(x)+√(8 - x)))/((√(x)-√(8 - x))\cdot (√(x)+√(8 - x)))`

Third, simplify and evaluate the resulting expression:

`\lim_(x \to 4) ((x - 4)\cdot (√(x)+√(8-x)))/(x - (8 - x))`

`\lim_(x \to 4) ((x - 4)\cdot (√(x) + √(8 - x)))/(8)`

0

By eliminating the evitable discontinuity, the limit of the rational function with radical elements is equal to 0 when x tends to 4.