Question - Finding Value of 'a' for Continuous Function at x = 2
Solution:
The function $$ f(x) $$ is defined by three expressions for different ranges of $$ x $$, and we are asked to find the value of "a" given that the function is continuous at $$ x = 2 $$.For a function to be continuous at a point, the left-hand limit (as $$ x $$ approaches the point from the left), the right-hand limit (as $$ x $$ approaches the point from the right), and the function's value at the point must all be the same.Let's evaluate the limit from the left ($$ x < 2 $$) and from the right ($$ x > 2 $$) as well as the function's value at $$ x = 2 $$:- The right-hand limit as $$ x $$ approaches 2 from the right is given by the expression $$ x + a $$. As $$ x $$ approaches 2, this expression becomes $$ 2 + a $$.- The function's value at $$ x = 2 $$ is given as 5.- The left-hand limit as $$ x $$ approaches 2 from the left is given by the expression $$ 2x + 1 $$. As $$ x $$ approaches 2, this expression becomes $$ 2(2) + 1 = 4 + 1 = 5 $$.Since the function is continuous at $$ x = 2 $$, the right-hand limit must equal the left-hand limit and also the function's value at $$ x = 2 $$.So we have the equation $$ 2 + a = 5 $$.Solving for $$ a $$:\[ a = 5 - 2 \]\[ a = 3 \]Therefore, the value of "a" that makes the function continuous at $$ x = 2 $$ is 3.
CamTutor
In regards to math, we are professionals.
Get In Touch
Email: camtutor.ai@gmail.com