4. Let $$ T(x)=\left\{\begin{array}{ll}\alpha x+4 & ext { if } x1 .\end{array} ight. $$ Find the value of $$ \alpha $$ so that $$ \lim _{x ightarrow 1} T(x) $$ exists.

Question

4. Let $$ T(x)=\left\{\begin{array}{ll}\alpha x+4 & 	ext { if } x<1 \ 4 & 	ext { if } x=1 \ x^{2}+1 & 	ext { if } x>1 .\end{array}ight. $$ Find the value of $$ \alpha $$ so that $$ \lim _{x ightarrow 1} T(x) $$ exists.

UpStudy AI · Accepted Answer

We need the limit

$$
\lim_{x \rightarrow 1} T(x)
$$

to exist.

**Step 1: Compute the left-hand limit.**

For $$ x < 1 $$, we have

$$
T(x) = \alpha x + 4.
$$

Thus, the left-hand limit as $$ x \to 1^- $$ is

$$
\lim_{x \to 1^-} T(x) = \alpha \cdot 1 + 4 = \alpha + 4.
$$

**Step 2: Compute the right-hand limit.**

For $$ x > 1 $$, we have

$$
T(x) = x^2 + 1.
$$

Thus, the right-hand limit as $$ x \to 1^+ $$ is

$$
\lim_{x \to 1^+} T(x) = 1^2 + 1 = 2.
$$

**Step 3: Set the left-hand and right-hand limits equal.**

For the overall limit to exist, we must have

$$
\alpha + 4 = 2.
$$

**Step 4: Solve for $$ \alpha $$.**

Subtract 4 from both sides:

$$
\alpha = 2 - 4 = -2.
$$

Hence, the value of $$ \alpha $$ is

$$
\alpha = -2.
$$
The value of $$ \alpha $$ is $$-2$$.

Let Find the value of so that exists.

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