Let a=8, f(x)=|x-8|/x
You are given lim x->a+ f(x)=lim x->8+
|x-8|/x
Is the limit of the value 0.
To determine the limit of the given expression, we need to consider the behavior of the function as x approaches the given value. In this case, the given value is a = 8.
To evaluate the limit lim x→8+ |x-8|/x, we can start by substituting x = 8 into the expression and see if it yields a specific value. However, directly substituting x = 8 gives us 0/8 which is undefined.
To proceed, we can simplify the expression by using the properties of absolute value. Recall that |x| = x when x ≥ 0, and |x| = -x when x < 0.
In the given expression |x-8|/x, we can observe that x-8 is always non-negative for x > 8, so |x-8| is simply equal to (x-8). Therefore, the expression simplifies to (x-8)/x.
Now, we can evaluate the limit lim x→8+ (x-8)/x by substituting x = 8 into the simplified expression. Doing so gives us (8-8)/8 = 0/8, which is still undefined.
Since directly substituting x = 8 did not yield a specific value for the limit, we need to employ a different approach. In this case, we can analyze the expression as x approaches 8 from the right side (x > 8) to determine the limit.
As x approaches 8 from the right side, the expression (x-8)/x will approach 0. This is because the numerator (x-8) approaches 0, while the denominator (x) remains positive and non-zero.
Thus, we can conclude that the limit lim x→8+ |x-8|/x = 0.