This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To evaluate the limit and discuss continuity, we will first simplify the function and then apply the conditions for continuity. Step 1: Evaluate the limit _x 1 (x^2 - 1)/(x - 1). When we substitute x=1 into the function, we get (1^2 - 1)/(1 - 1) = (0)/(0), which is an indeterminate form. We can simplify the expression by factoring the numerator. The numerator x^2 - 1 is a difference of squares, which factors as (x-1)(x+1). _x 1 (x^2 - 1)/(x - 1) = _x 1 ((x-1)(x+1))/(x - 1) For x ≠ 1, we can cancel the (x-1) term: _x 1 (x+1) Now, substitute x=1: 1+1 = 2 So, the limit is 2. Step 2: Discuss the continuity of the function at x=1. For a function f(x) to be continuous at a point x=c, three conditions must be met: 1. f(c) must be defined. 2. _x c f(x) must exist. 3. _x c f(x) = f(c). Let's check these conditions for f(x) = (x^2 - 1)/(x - 1) at x=1: 1. Is f(1) defined? Substituting x=1 into the original function gives f(1) = (1^2 - 1)/(1 - 1) = (0)/(0), which is undefined. 2. Does _x 1 f(x) exist? From Step 1, we found that _x 1 f(x) = 2. So, the limit exists. 3. Is _x 1 f(x) = f(1)? Since f(1) is undefined, this condition cannot be satisfied. Because f(1) is undefined, the function is not continuous at x=1. This type of discontinuity is called a removable discontinuity because the limit exists. The limit is 2. The function is not continuous at x=1 because f(1) is undefined. That's 2 down. 3 left today — send the next one.