Find the values of a and b that make f continuous...

Find the values of $a$ and $b$ that make $f$ continuous everywhere.\ $f(x) = egin{cases} frac{x^2 - 4}{x - 2} & ext{if } x < 2\ ax^2 - bx + 3 & ext{if } 2 le x < 3\ 2x - a + b & ext{if } x ge 3 end{cases}$

Transcript

00:02 Okay, showing this question, we have a piecewise function and it is given that the function is continuous on the interval of minus infinite to infinite.

00:13 Right, and we need to find the values of a and b.

00:17 So first of all, we need to simplify the first function, that is x square minus 4 divided by x minus 2.

00:25 So for the numerator, we can write x plus 2 multiplied by x minus 2 and denominator will be same.

00:33 Right now here x minus 2 will be cancel out so we can write only x plus 2 right so now for the first function we can just write x plus 2 right okay now here the function is continuous at everywhere right which means the left -hand side limit and right and right and side limit of the function as x tends to 2 that will be equal right so here we can write limit x tends to 2 minus f of x that should be equal to limit x tends to 2 plus f of x right now for the left side limit we need to choose the our first function and for the right -hand side limit we need to choose the our second function now execute the limit by substituting x is equal to 2 now simplify this right so here 4 minus 3 that is equal to 1 right now here we can interchange these two terms so we can write 2b that is equal to 4a minus 1 so we can write b is equal to 2a minus 1 by 2 now here the here the given function is continuous at everywhere so we can write here limit x tends to 3 minus f of x that should be equal to limit x 2 3 plus f of x right which means the left -hand side and right -hand side limit of the function as x -tenths to 3 should be equal right so for the left -hand side limit we need to choose the second equation and for the right -hand side limit we need to choose our last equation that is 2x minus a plus b right now execute the limit by substituting x is equal to 3 now simplify this...

Question

Find the values of $a$ and $b$ that make $f$ continuous everywhere.\ $f(x) = egin{cases} frac{x^2 - 4}{x - 2} & ext{if } x < 2\ ax^2 - bx + 3 & ext{if } 2 le x < 3\ 2x - a + b & ext{if } x ge 3 end{cases}$

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Find the values of $a$ and $b$ that make $f$ continuous everywhere.\ $f(x) = egin{cases} frac{x^2 - 4}{x - 2} & ext{if } x < 2\ ax^2 - bx + 3 & ext{if } 2 le x < 3\ 2x - a + b & ext{if } x ge 3 end{cases}$