Evaluate the limit, if it exists. (If an answer does not exist, enter DNE.) lim x→−24 x2 + 49 − 25x + 24

To evaluate the limit

$\lim_{x \to -24} \left(x^2 + 49 - 25x + 24\right),$

we will follow the steps outlined:

1. Break Down the Problem

We will simplify the expression $x^2 + 49 - 25x + 24$ and then substitute $x = -24$ to find the limit.

2. Relevant Concepts

To evaluate the limit, we need to plug in $x = -24$ into the simplified expression.

3. Analysis and Detail

Start with the expression:

$x^2 - 25x + 49 + 24.$

Combining constants:

$x^2 - 25x + 73.$

Now, substitute $x = -24$:

$(-24)^2 - 25(-24) + 73.$

Calculating each term:

1. $(-24)^2 = 576$,
2. $-25 \times -24 = 600$,
3. Adding these and the constant $73$:

$576 + 600 + 73.$

Calculating step-by-step:

• First, $576 + 600 = 1176$,
• Then, $1176 + 73 = 1249$.

4. Verify and Summarize

We calculated the limit by evaluation directly through substitution and ensured each calculation was conducted step-by-step.

Final Answer

Thus, the limit exists and is given by:

$\lim_{x \to -24} \left(x^2 + 49 - 25x + 24\right) = 1249.$