Evaluate the following limits, if they exist :
`lim_(x to 0)((x-1)^(2)+5)`

Options

A. 6
B. 5
C. 4
D. 3

Answer

To evaluate the limit \( \lim_{x \to 0} ((x-1)^2 + 5) \), we will follow these steps:

### Step 1: Substitute \( x = 0 \) into the expression
We start by substituting \( x = 0 \) into the expression \( (x-1)^2 + 5 \).

\[ (x-1)^2 + 5 = (0-1)^2 + 5 \]

### Step 2: Simplify the expression
Now we simplify \( (0-1)^2 + 5 \):

\[ (0-1)^2 = (-1)^2 = 1 \]

So, we have:

\[ 1 + 5 = 6 \]

### Step 3: Conclusion
Thus, the limit is:

\[ \lim_{x \to 0} ((x-1)^2 + 5) = 6 \]

### Final Answer
The limit exists and is equal to \( 6 \).

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