Questions: If (f(x)=x^2-2 x-3, x neq 2; k-3, x=2) find (k) such that (lim x rightarrow 2 f(x)=f(2))

Transcript text: If $f(x)=\left\{\begin{array}{ll}x^{2}-2 x-3, & x \neq 2 \\ k-3, & x=2\end{array}\right.$ find $k$ such that $\lim _{x \rightarrow 2} f(x)=f(2)$

Solution

Solution Steps

To find the value of $k$ such that $\lim _{x \rightarrow 2} f(x)=f(2)$, we need to evaluate the limit of $f(x)$ as $x$ approaches 2 and compare it to the value of $f(2)$.

Calculate the limit of $f(x)$ as $x$ approaches 2 by substituting $x=2$ into the function $f(x)$.
Calculate the value of $f(2)$ by substituting $x=2$ into the function $f(x)$.
Set the limit equal to $f(2)$ and solve for $k$ to find the value of $k$.

Step 1: Find the limit as x approaches 2

To find the limit as x approaches 2, we substitute x=2 into the function: $f(2) = 2^2 - 2(2) - 3 = 4 - 4 - 3 = -3$

Step 2: Set up the limit equation

The limit as x approaches 2 is given by: $\lim_{x \to 2} f(x) = f(2)$

Step 3: Find the limit as x approaches 2

Since the limit is equal to $f(2)$, we have: $\lim_{x \to 2} f(x) = -3$