Using the graph, find the specified limit or state that the limit does not exist (DNE).limx→0−f(x)\lim_{x\rightarrow0^{-}}f\left(x\right)limx→0−f(x) , limx→0+f(x)\lim_{x\rightarrow0^{+}}f\left(x\right)limx→0+f(x), limx→0f(x)\lim_{x\rightarrow0}f\left(x\right)limx→0f(x)

lim ⁡ x → 0 − f ( x ) = 0 \lim_{x\rightarrow0^{-}}f\left(x\right)=0 lim x → 0 − f ( x ) = 0 , lim ⁡ x → 0 + f ( x ) = 0 \lim_{x\rightarrow0^{+}}f\left(x\right)=0 lim x → 0 + f ( x ) = 0 , lim ⁡ x → 0 f ( x ) = 0 \lim_{x\rightarrow0}f\left(x\right)=0 lim x → 0 f ( x ) = 0

Using the graph, find the specified limit or state that the limit does not exist (DNE). lim x → 0 − f ( x ) \lim_{x\rightarrow0^{-}}f\left(x\right) lim x → 0 − f ( x ) , lim x → 0 + f ( x ) \lim_{x\rightarrow0^{+}}f\left(x\right) lim x → 0 + f ( x ) , lim x → 0 f ( x ) \lim_{x\rightarrow0}f\left(x\right) lim x → 0 f ( x )

Using the graph given to us here, we want to find the specified limits or otherwise state that it does not exist. Now our first limit here is a left sided limit, the limit of f of x as x approaches 0 from the left. Now coming in from that left side approaching that value of 0, I see that my function is coming close to a value of 0 as well. Now from that right side, my right sided limit as x approaches 0 from the right, I see that again my function is kind of closing in and getting really close to this value of 0. Now since these two limits, my one-sided limits from the left and the right, are equal to each other, I already know what my limit of f of x as x approaches 0 is. It's that same value of 0. Now again here, reminder that it does not matter what the value of our function is at 0, just what our function is doing as x gets really, really close to 0. Thanks for watching, and I'll see you in the next one.

Using the graph, find the specified limit or state that the limit does not exist (DNE). lim ⁡ x → 0 − f ( x ) \lim_{x\rightarrow0^{-}}f\left(x\right) lim x → 0 − f ( x ) , lim ⁡ x → 0 + f ( x ) \lim_{x\rightarrow0^{+}}f\left(x\right) lim x → 0 + f ( x ) , lim ⁡ x → 0 f ( x ) \lim_{x\rightarrow0}f\left(x\right) lim x → 0 f ( x )

lim ⁡ x → 0 − f ( x ) = 0 \lim_{x\rightarrow0^{-}}f\left(x\right)=0 lim x → 0 − f ( x ) = 0 , lim ⁡ x → 0 + f ( x ) = 0 \lim_{x\rightarrow0^{+}}f\left(x\right)=0 lim x → 0 + f ( x ) = 0 , lim ⁡ x → 0 f ( x ) = 0 \lim_{x\rightarrow0}f\left(x\right)=0 lim x → 0 f ( x ) = 0

lim ⁡ x → 0 − f ( x ) = 0 \lim_{x\rightarrow0^{-}}f\left(x\right)=0 lim x → 0 − f ( x ) = 0 , lim ⁡ x → 0 + f ( x ) = 0 \lim_{x\rightarrow0^{+}}f\left(x\right)=0 lim x → 0 + f ( x ) = 0 , lim ⁡ x → 0 f ( x ) = D N E \lim_{x\rightarrow0}f\left(x\right)=DNE lim x → 0 f ( x ) = D NE

lim ⁡ x → 0 − f ( x ) = − 1 \lim_{x\rightarrow0^{-}}f\left(x\right)=-1 lim x → 0 − f ( x ) = − 1 , lim ⁡ x → 0 + f ( x ) = − 1 \lim_{x\rightarrow0^{+}}f\left(x\right)=-1 lim x → 0 + f ( x ) = − 1 , lim ⁡ x → 0 f ( x ) = D N E \lim_{x\rightarrow0}f\left(x\right)=DNE lim x → 0 f ( x ) = D NE

lim ⁡ x → 0 − f ( x ) = − 1 \lim_{x\rightarrow0^{-}}f\left(x\right)=-1 lim x → 0 − f ( x ) = − 1 , lim ⁡ x → 0 + f ( x ) = − 1 \lim_{x\rightarrow0^{+}}f\left(x\right)=-1 lim x → 0 + f ( x ) = − 1 , lim ⁡ x → 0 f ( x ) = − 1 \lim_{x\rightarrow0}f\left(x\right)=-1 lim x → 0 f ( x ) = − 1

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).
$$\lim$_{x$\rightarrow$0^{-}}f$\left$(x$\right$)$ , $$\lim$_{x$\rightarrow$0^{+}}f$\left$(x$\right$)$ , $$\lim$_{x$\rightarrow$0}f$\left$(x$\right$)$

$$\lim$_{x$\rightarrow$0^{-}}f$\left$(x$\right$)=0$ , $$\lim$_{x$\rightarrow$0^{+}}f$\left$(x$\right$)=0$ , $$\lim$_{x$\rightarrow$0}f$\left$(x$\right$)=0$

$$\lim$_{x$\rightarrow$0^{-}}f$\left$(x$\right$)=0$ , $$\lim$_{x$\rightarrow$0^{+}}f$\left$(x$\right$)=0$ , $$\lim$_{x$\rightarrow$0}f$\left$(x$\right$)=DNE$

$$\lim$_{x$\rightarrow$0^{-}}f$\left$(x$\right$)=-1$ , $$\lim$_{x$\rightarrow$0^{+}}f$\left$(x$\right$)=-1$ , $$\lim$_{x$\rightarrow$0}f$\left$(x$\right$)=DNE$

$$\lim$_{x$\rightarrow$0^{-}}f$\left$(x$\right$)=-1$ , $$\lim$_{x$\rightarrow$0^{+}}f$\left$(x$\right$)=-1$ , $$\lim$_{x$\rightarrow$0}f$\left$(x$\right$)=-1$

Verified step by step guidance

Step 1: Observe the graph of f(x) and focus on the behavior of the function as x approaches 0 from the left (x → 0⁻). The graph shows that as x approaches 0 from the left, the value of f(x) approaches -1. This indicates lim_{x→0⁻}f(x) = -1.

Step 2: Next, examine the behavior of f(x) as x approaches 0 from the right (x → 0⁺). The graph shows that as x approaches 0 from the right, the value of f(x) also approaches -1. This indicates lim_{x→0⁺}f(x) = -1.

Step 3: To determine lim_{x→0}f(x), check if the left-hand limit (lim_{x→0⁻}f(x)) and the right-hand limit (lim_{x→0⁺}f(x)) are equal. Since both limits are equal to -1, the two-sided limit exists and is equal to -1. Therefore, lim_{x→0}f(x) = -1.

Step 4: Summarize the results: lim_{x→0⁻}f(x) = -1, lim_{x→0⁺}f(x) = -1, and lim_{x→0}f(x) = -1.

Step 5: Ensure clarity by verifying the graph and confirming that there are no discontinuities or jumps at x = 0 that would contradict the conclusion.

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