Questions: Find the limit [ lim x rightarrow-2 fracx+2x^2+9 x+14= ]

$Find the limit [ lim x rightarrow-2 fracx+2x^2+9 x+14= ]$

Transcript text: Find the limit \[ \lim _{x \rightarrow-2} \frac{x+2}{x^{2}+9 x+14}= \]

Solution

Solution Steps

To find the limit of the given function as $ x $ approaches $-2$, we first check if direct substitution results in an indeterminate form. If it does, we attempt to simplify the expression, often by factoring, to resolve the indeterminate form. Once simplified, we substitute $-2$ into the expression to find the limit.

Step 1: Evaluate the Limit

We need to find the limit \[ \lim_{x \to -2} \frac{x + 2}{x^2 + 9x + 14}. \] First, we substitute $ x = -2 $ into the expression. This gives us: \[ \frac{-2 + 2}{(-2)^2 + 9(-2) + 14} = \frac{0}{4 - 18 + 14} = \frac{0}{0}, \] which is an indeterminate form.

Step 2: Simplify the Expression

To resolve the indeterminate form, we factor the denominator. The quadratic $ x^2 + 9x + 14 $ can be factored as: \[ x^2 + 9x + 14 = (x + 2)(x + 7). \] Thus, we can rewrite the limit as: \[ \lim_{x \to -2} \frac{x + 2}{(x + 2)(x + 7)}. \] We can cancel the $ x + 2 $ terms (valid since we are considering the limit and not the value at $ x = -2 $): \[ \lim_{x \to -2} \frac{1}{x + 7}. \]

Step 3: Substitute and Calculate the Limit

Now we substitute $ x = -2 $ into the simplified expression: \[ \frac{1}{-2 + 7} = \frac{1}{5}. \]