Using the graph, find the specified limit or state that the limit does not exist.limx→4−f(x)\lim_{x\rightarrow4^{-}}f\left(x\right)limx→4−f(x), limx→4+f(x)\lim_{x\rightarrow4^{+}}f\left(x\right)limx→4+f(x), limx→4f(x)\lim_{x\rightarrow4}f\left(x\right)limx→4f(x)

lim ⁡ x → 4 − f ( x ) = 1 \lim_{x\rightarrow4^{-}}f\left(x\right)=1 lim x → 4 − f ( x ) = 1 , lim ⁡ x → 4 + f ( x ) = 3 \lim_{x\rightarrow4^{+}}f\left(x\right)=3 lim x → 4 + f ( x ) = 3 , lim ⁡ x → 4 f ( x ) = D N E \lim_{x\rightarrow4}f\left(x\right)=DNE lim x → 4 f ( x ) = D NE

Using the graph, find the specified limit or state that the limit does not exist. lim x → 4 − f ( x ) \lim_{x\rightarrow4^{-}}f\left(x\right) lim x → 4 − f ( x ) , lim x → 4 + f ( x ) \lim_{x\rightarrow4^{+}}f\left(x\right) lim x → 4 + f ( x ) , lim x → 4 f ( x ) \lim_{x\rightarrow4}f\left(x\right) lim x → 4 f ( x )

Using the graph shown to us here, we wanna find the specified limit or otherwise state that it does not exist. So let's jump in here to finding our left sided limit, the limit of f of x as x approaches 4 from the left. Remember that that's how we read that limit notation here with that little negative sign. So looking at my graph here, as x approaches 4 from the left side here, I see that my function is getting close to a value of 1. So here, that tells me that my limit, that left sided limit, is equal to 1. Now coming in from the right side here, my right sided limit as x approaches 4 from the right, I see that my function is instead approaching a value of 3. So that right sided limit is equal to 3. Now looking at my final limit that I'm asked to find here, the limit of f of x as x approaches 4, we already found our left sided and right sided limits. And looking at these two values, they are completely different. My left side limit is 1. My right sided limit is 3. Now remember that if our one-sided limits are not the same, then our limit, the limit of f of x as x approaches 4, simply does not exist. So we can't say that there is a limit here because they are completely different coming in from either side. Now something that you might be thinking is looking at our function. Wouldn't it just be our function value of 1? Because that's actually where it's defined at 4. But no. Remember that it does not matter what the value of our function is at 4, just what it's doing as it gets really, really close to 4 from either side. So So if it's not doing the same thing from either side, then this limit simply does not exist. Let me know if you have any questions here, but I'll see you in the next video.

Using the graph, find the specified limit or state that the limit does not exist. lim ⁡ x → 4 − f ( x ) \lim_{x\rightarrow4^{-}}f\left(x\right) lim x → 4 − f ( x ) , lim ⁡ x → 4 + f ( x ) \lim_{x\rightarrow4^{+}}f\left(x\right) lim x → 4 + f ( x ) , lim ⁡ x → 4 f ( x ) \lim_{x\rightarrow4}f\left(x\right) lim x → 4 f ( x )

lim ⁡ x → 4 − f ( x ) = 1 \lim_{x\rightarrow4^{-}}f\left(x\right)=1 lim x → 4 − f ( x ) = 1 , lim ⁡ x → 4 + f ( x ) = 3 \lim_{x\rightarrow4^{+}}f\left(x\right)=3 lim x → 4 + f ( x ) = 3 , lim ⁡ x → 4 f ( x ) = D N E \lim_{x\rightarrow4}f\left(x\right)=DNE lim x → 4 f ( x ) = D NE

lim ⁡ x → 4 − f ( x ) = 1 \lim_{x\rightarrow4^{-}}f\left(x\right)=1 lim x → 4 − f ( x ) = 1 , lim ⁡ x → 4 + f ( x ) = 1 \lim_{x\rightarrow4^{+}}f\left(x\right)=1 lim x → 4 + f ( x ) = 1 , lim ⁡ x → 4 f ( x ) = 1 \lim_{x\rightarrow4}f\left(x\right)=1 lim x → 4 f ( x ) = 1

lim ⁡ x → 4 − f ( x ) = 3 \lim_{x\rightarrow4^{-}}f\left(x\right)=3 lim x → 4 − f ( x ) = 3 , lim ⁡ x → 4 + f ( x ) = 3 \lim_{x\rightarrow4^{+}}f\left(x\right)=3 lim x → 4 + f ( x ) = 3 , lim ⁡ x → 4 f ( x ) = 3 \lim_{x\rightarrow4}f\left(x\right)=3 lim x → 4 f ( x ) = 3

lim ⁡ x → 4 − f ( x ) = 3 \lim_{x\rightarrow4^{-}}f\left(x\right)=3 lim x → 4 − f ( x ) = 3 , lim ⁡ x → 4 + f ( x ) = 1 \lim_{x\rightarrow4^{+}}f\left(x\right)=1 lim x → 4 + f ( x ) = 1 , lim ⁡ x → 4 f ( x ) = D N E \lim_{x\rightarrow4}f\left(x\right)=DNE lim x → 4 f ( x ) = D NE

1. Limits and Continuity

Introduction to Limits

Business Calculus

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Introduction to Limits

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Multiple Choice

Using the graph, find the specified limit or state that the limit does not exist.
$$\lim$_{x$\rightarrow$4^{-}}f$\left$(x$\right$)$ , $$\lim$_{x$\rightarrow$4^{+}}f$\left$(x$\right$)$ , $$\lim$_{x$\rightarrow$4}f$\left$(x$\right$)$

$$\lim$_{x$\rightarrow$4^{-}}f$\left$(x$\right$)=1$ , $$\lim$_{x$\rightarrow$4^{+}}f$\left$(x$\right$)=1$ , $$\lim$_{x$\rightarrow$4}f$\left$(x$\right$)=1$

$$\lim$_{x$\rightarrow$4^{-}}f$\left$(x$\right$)=3$ , $$\lim$_{x$\rightarrow$4^{+}}f$\left$(x$\right$)=3$ , $$\lim$_{x$\rightarrow$4}f$\left$(x$\right$)=3$

$$\lim$_{x$\rightarrow$4^{-}}f$\left$(x$\right$)=3$ , $$\lim$_{x$\rightarrow$4^{+}}f$\left$(x$\right$)=1$ , $$\lim$_{x$\rightarrow$4}f$\left$(x$\right$)=DNE$

$$\lim$_{x$\rightarrow$4^{-}}f$\left$(x$\right$)=1$ , $$\lim$_{x$\rightarrow$4^{+}}f$\left$(x$\right$)=3$ , $$\lim$_{x$\rightarrow$4}f$\left$(x$\right$)=DNE$

Verified step by step guidance

Step 1: Observe the graph of f(x) and identify the behavior of the function as x approaches 4 from the left (x → 4⁻). The graph shows that as x approaches 4 from the left, the function value approaches 1. Therefore, lim_{x→4⁻}f(x) = 1.

Step 2: Next, examine the graph as x approaches 4 from the right (x → 4⁺). The graph indicates that as x approaches 4 from the right, the function value approaches 3. Therefore, lim_{x→4⁺}f(x) = 3.

Step 3: Determine the overall limit as x approaches 4 (x → 4). Since the left-hand limit (lim_{x→4⁻}f(x)) is 1 and the right-hand limit (lim_{x→4⁺}f(x)) is 3, the two limits are not equal. Therefore, the limit does not exist (DNE).

Step 4: Summarize the findings: lim_{x→4⁻}f(x) = 1, lim_{x→4⁺}f(x) = 3, and lim_{x→4}f(x) = DNE.

Step 5: Understand the concept: A limit exists at a point only if the left-hand limit and right-hand limit are equal. If they differ, the overall limit does not exist.

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