Questions: Given lim as x approaches 11 of f(x)=-5 and lim as x approaches 11 of g(x)=-12, evaluate lim as x approaches 11 of [5f(x)-2g(x)]. lim as u approaches +1 of [5f(x)-2g(x)]=-1 lim as u approaches ++ of [5f(x)-2g(x)]=-49 lim as v approaches . of [5f(x)-2g(x)]=1 lim as x approaches +... of [5f(x)-2g(x)]=49

Given lim as x approaches 11 of f(x)=-5 and lim as x approaches 11 of g(x)=-12, evaluate lim as x approaches 11 of [5f(x)-2g(x)].
lim as u approaches +1 of [5f(x)-2g(x)]=-1
lim as u approaches ++ of [5f(x)-2g(x)]=-49
lim as v approaches . of [5f(x)-2g(x)]=1
lim as x approaches +... of [5f(x)-2g(x)]=49

Transcript text: Given $\lim _{x \rightarrow 11} f(x)=-5$ and $\lim _{x \rightarrow 11} g(x)=-12$, evaluate $\lim _{x \rightarrow 11}[5 f(x)-2 g(x)]$. $\lim _{u \rightarrow+1}[5 f(x)-2 g(x)]=-1$ $\lim _{u \rightarrow+{ }^{+}}[5 f(x)-2 g(x)]=-49$ $\lim _{v \rightarrow \cdot}[5 f(x)-2 g(x)]=1$ $\lim _{x \rightarrow+\cdots}[5 f(x)-2 g(x)]=49$

Solution

Solution Steps

Step 1: Understand the given limits

We are given: \[ \lim_{x \rightarrow 11} f(x) = -5 \quad \text{and} \quad \lim_{x \rightarrow 11} g(x) = -12. \]

Step 2: Apply the limit properties

The limit of a linear combination of functions is the linear combination of their limits. Therefore: \[ \lim_{x \rightarrow 11} [5f(x) - 2g(x)] = 5 \cdot \lim_{x \rightarrow 11} f(x) - 2 \cdot \lim_{x \rightarrow 11} g(x). \]

Step 3: Substitute the given limits

Substitute the given values of $\lim_{x \rightarrow 11} f(x)$ and $\lim_{x \rightarrow 11} g(x)$ into the equation: \[ \lim_{x \rightarrow 11} [5f(x) - 2g(x)] = 5 \cdot (-5) - 2 \cdot (-12). \]

Step 4: Perform the calculations

Calculate the expression: \[ 5 \cdot (-5) = -25 \quad \text{and} \quad -2 \cdot (-12) = 24. \] Now, add the results: \[ -25 + 24 = -1. \]