Evaluate the limit $\lim_{x\to2}\left(\frac{x^2-x+5}{x-2}\right)$ by replacing all occurrences of $x$ by $2$

Subtract the values $2$ and $-2$

Subtract the values $5$ and $-2$

Calculate the power $2^2$

Add the values $4$ and $3$

An expression divided by zero tends to infinity

As by directly replacing the value to which the limit tends, we obtain an indeterminate form, we must try replacing a value close but not equal to $2$. In this case, since we are approaching $2$ from the left, let's try replacing a slightly smaller value, such as $1.99999$ in the function within the limit:

As by directly replacing the value to which the limit tends, we obtain an indeterminate form, we must try replacing a value close but not equal to $2$. In this case, since we are approaching $2$ from the right, let's try replacing a slightly larger value, such as $2.00001$ in the function within the limit:

Once we have found both limits from the left side and from the right side, we check if they are both the same for the limit to exist. Since $\lim_{x\to c^+}f(x) \neq \lim_{x\to c^-}f(x)$, then the limit does not exist