(2) (optional; do not submit) Consider the function: f(x) =...

(2) (optional; do not submit) Consider the function: $f(x) = \sqrt{\frac{8}{9x^2 - 1}}$ (a) Use the $\epsilon-\delta$ language to prove $f$ is continuous at $x = 1$ (b) Use the Sequential Criterion and limit rules to prove $f$ is continuous at $x = 1$ $\lim_{x \to 1} = 1.$

Transcript

00:01 The given function is f of x is equal to 5x minus 2 x not equal to 1 and negative 2 when x is equal to 1.

00:18 Let's find the limit limit extending to 1 f of x find the limit when extends to 1 f of x we need to find both left and right hand limits so limit x 10 into 1 plus f of x so what do you mean by 1 plus approaching 1 from the right so when it approaches from 1 from the right i need to select this particular function because x is not equal to 1 remember x is approaching 1 from the right but x can never be equal to 1 so that's why limit x 10 into 1 plus f of x is 5x minus 2 and that is 3 whereas left -hand limit limit extending to 1 minus f of x 1 minus is it is approaching 1 from the left but x can never be 1 again you should take this same thing so it's limit x tending to 1 minus 5 x minus 2 again it is approaching 3 so both left and right hand limits are same so limit exists and the limit is equal to 3 so the answer is 3 the same part a let's do it by epsilon delta method so what does epsilon delta method represents what is the definition of limit using a delta for all it's not positive there exists a delta such that mod x minus a less than delta implies mod f of x minus f of a is less than epsilon then we say limit x ten into a f of x is sorry not f of a l is l right f of a is for continuity i'm sorry for that so it's l so we say that limit extending to a f of x is equal to l now let's do this so what we need to do here is we need to start something like this the the proof construct the proof structure is something like this we start with mod f of x minus we need to prove that the limit is 3 because we already got the limit as 3 so we need to prove that it is 3 so we try to take this mod f of x minus 3 and we'll start with mod f of x minus 3 is less than epsilon but what is f of x it is 5x minus 2 minus 3 is less than epsilon that implies mod 5x minus 5 is less than epsilon that implies mod x minus 1 is less than epsilon by 5 so that means we can choose our delta to be epsilon by 5 so choose delta to be epsilon by 5 because delta is a function of epsilon so delta is some g of epsilon.

03:12 Now let's start the actual proof...

Question

(2) (optional; do not submit) Consider the function: $f(x) = \sqrt{\frac{8}{9x^2 - 1}}$ (a) Use the $\epsilon-\delta$ language to prove $f$ is continuous at $x = 1$ (b) Use the Sequential Criterion and limit rules to prove $f$ is continuous at $x = 1$ $\lim_{x \to 1} = 1.$

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