Using the graph, find the specified limit or state that the limit does not exist (DNE).limx→−2−f(x)\lim_{x\rarr-2^{-}}f\left(x\right)limx→−2−f(x), limx→−2+f(x)\lim_{x\rarr-2^{+}}f\left(x\right)limx→−2+f(x), limx→−2f(x)\lim_{x\rarr-2^{}}f\left(x\right)limx→−2f(x)

lim ⁡ x → − 2 − f ( x ) = 1 \lim_{x\rarr-2^{-}}f\left(x\right)=1 lim x → − 2 − f ( x ) = 1 , lim ⁡ x → − 2 + f ( x ) = 1 \lim_{x\rarr-2^{+}}f\left(x\right)=1 lim x → − 2 + f ( x ) = 1 , lim ⁡ x → − 2 f ( x ) = 1 \lim_{x\rarr-2^{}}f\left(x\right)=1 lim x → − 2 f ( x ) = 1

Using the graph, find the specified limit or state that the limit does not exist (DNE). lim x → − 2 − f ( x ) \lim_{x\rarr-2^{-}}f\left(x\right) lim x → − 2 − f ( x ) , lim x → − 2 + f ( x ) \lim_{x\rarr-2^{+}}f\left(x\right) lim x → − 2 + f ( x ) , lim x → − 2 f ( x ) \lim_{x\rarr-2^{}}f\left(x\right) lim x → − 2 f ( x )

Using the graph, we wanna go ahead and find the specified limits or state that they don't exist. So let's go ahead and jump in here. Now the first limit that we're asked to find is a left sided limit, the limit of f of x as x approaches negative 2 from the left. So looking at our graph here, x approaching negative 2 coming in from this left side, I see that as x gets really, really close to negative 2 from the left, that my function is approaching a value of 1. So that's my limit there. Now looking next at my right sided limit, the limit of f of x as x approaches negative 2 from the right, Coming in now from that right side, approaching that value of negative 2, getting really, really close to it from the other side, I see again that my function is approaching a value of 1. Now since we found those one-sided limits, what's just our limit as x approaches a negative 2? Now since these limits are equal to each other, my left sided limit is equal to my right sided limit as x approaches negative 2. That tells me that my limit of f of x as x approaches negative 2 is simply equal to that same value 1. Now I'll see you in the next one. Thanks for watching.

Using the graph, find the specified limit or state that the limit does not exist (DNE). lim ⁡ x → − 2 − f ( x ) \lim_{x\rarr-2^{-}}f\left(x\right) lim x → − 2 − f ( x ) , lim ⁡ x → − 2 + f ( x ) \lim_{x\rarr-2^{+}}f\left(x\right) lim x → − 2 + f ( x ) , lim ⁡ x → − 2 f ( x ) \lim_{x\rarr-2^{}}f\left(x\right) lim x → − 2 f ( x )

lim ⁡ x → − 2 − f ( x ) = 1 \lim_{x\rarr-2^{-}}f\left(x\right)=1 lim x → − 2 − f ( x ) = 1 , lim ⁡ x → − 2 + f ( x ) = 1 \lim_{x\rarr-2^{+}}f\left(x\right)=1 lim x → − 2 + f ( x ) = 1 , lim ⁡ x → − 2 f ( x ) = 1 \lim_{x\rarr-2^{}}f\left(x\right)=1 lim x → − 2 f ( x ) = 1

lim ⁡ x → − 2 − f ( x ) = 1 \lim_{x\rarr-2^{-}}f\left(x\right)=1 lim x → − 2 − f ( x ) = 1 , lim ⁡ x → − 2 + f ( x ) = − 1 \lim_{x\rarr-2^{+}}f\left(x\right)=-1 lim x → − 2 + f ( x ) = − 1 , lim ⁡ x → − 2 f ( x ) = D N E \lim_{x\rarr-2}f\left(x\right)=DNE lim x → − 2 f ( x ) = D NE

lim ⁡ x → − 2 − f ( x ) = 1 \lim_{x\rarr-2^{-}}f\left(x\right)=1 lim x → − 2 − f ( x ) = 1 , lim ⁡ x → − 2 + f ( x ) = 1 \lim_{x\rarr-2^{+}}f\left(x\right)=1 lim x → − 2 + f ( x ) = 1 , lim ⁡ x → − 2 f ( x ) = D N E \lim_{x\rarr-2}f\left(x\right)=DNE lim x → − 2 f ( x ) = D NE

lim ⁡ x → − 2 − f ( x ) = 0 \lim_{x\rarr-2^{-}}f\left(x\right)=0 lim x → − 2 − f ( x ) = 0 , lim ⁡ x → − 2 + f ( x ) = 0 \lim_{x\rarr-2^{+}}f\left(x\right)=0 lim x → − 2 + f ( x ) = 0 , lim ⁡ x → − 2 f ( x ) = 0 \lim_{x\rarr-2}f\left(x\right)=0 lim x → − 2 f ( x ) = 0

1. Limits and Continuity

Introduction to Limits

Business Calculus

1. Limits and Continuity

Introduction to Limits

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Multiple Choice

Using the graph, find the specified limit or state that the limit does not exist (DNE).
$\lim_{x\rarr-2^{-}}f\left(x\right)$ , $\lim_{x\rarr-2^{+}}f\left(x\right)$ , $\lim_{x\rarr-2^{}}f\left(x\right)$

$\lim_{x\rarr-2^{-}}f\left(x\right)=1$ , $\lim_{x\rarr-2^{+}}f\left(x\right)=1$ , $\lim_{x\rarr-2^{}}f\left(x\right)=1$

$\lim_{x\rarr-2^{-}}f\left(x\right)=1$ , $\lim_{x\rarr-2^{+}}f\left(x\right)=-1$ , $\lim_{x\rarr-2}f\left(x\right)=DNE$

$\lim_{x\rarr-2^{-}}f\left(x\right)=1$ , $\lim_{x\rarr-2^{+}}f\left(x\right)=1$ , $\lim_{x\rarr-2}f\left(x\right)=DNE$

$\lim_{x\rarr-2^{-}}f\left(x\right)=0$ , $\lim_{x\rarr-2^{+}}f\left(x\right)=0$ , $\lim_{x\rarr-2}f\left(x\right)=0$

Verified step by step guidance

Step 1: Observe the graph of f(x) near x = -2. The graph shows the behavior of the function as x approaches -2 from the left (x → -2⁻) and from the right (x → -2⁺).

Step 2: To find lim⁡x→−2⁻f(x), examine the values of f(x) as x approaches -2 from the left side. Follow the curve to see where it is heading as x gets closer to -2 from the left.

Step 3: To find lim⁡x→−2⁺f(x), examine the values of f(x) as x approaches -2 from the right side. Follow the curve to see where it is heading as x gets closer to -2 from the right.

Step 4: To determine lim⁡x→−2f(x), check if the left-hand limit (lim⁡x→−2⁻f(x)) and the right-hand limit (lim⁡x→−2⁺f(x)) are equal. If they are not equal, the limit does not exist (DNE).

Step 5: Based on the graph, note that lim⁡x→−2⁻f(x) = 1, lim⁡x→−2⁺f(x) = -1, and since the left-hand limit and right-hand limit are not equal, lim⁡x→−2f(x) = DNE.

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