Q. 2 TF - Calculus

Question

State the limit-definition of continuity with a formal $\delta -\epsilon$statement.

Step-by-Step Solution

Answer

If limit $\underset{x\to c}{\lim }f\left(x\right)=L$ is continues then   $\delta >0$will exist for all $\epsilon >0$such that if  $0<|x-c|<\delta , \mathrm{then} \left|f\right(x)-L|<\epsilon .$

Step 1: Identify the Limit

We need to evaluate: \(State the limit-definition of continuity with a formal δ-ε statement.\).

Step 2: Analyze the Form

We check whether direct substitution works or if we have an indeterminate form that requires algebraic manipulation, L'Hopital's rule, or other techniques.

Step 3: Evaluate the Limit

Applying the appropriate technique, we evaluate the limit.

Step 4: State the Result

If limit $\underset{x\to c}{\lim }f\left(x\right)=L$ is continues then   $\delta >0$will exist for all $\epsilon >0$such that if  $0<|x-c|<\delta , \mathrm{then} \left|f\right(x)-L|<\epsilon .$