Evaluate the following limit: lim x → ∞ − 4 x + 2 \lim_{x\rarr\infty}-\frac{4}{x}+2 .

this problem, we're asked to evaluate the following limit, where we have the limit as x approaches infinity for negative four over x plus 2. Now to solve this problem, we need to first find any x's we have in this problem, which I can see we have an x right there in the denominator of this fraction. Now what I'm going to do is focus on this term specifically that has the x in it. We have negative 4 over x. And if I plug in a bunch of numbers for x here where the numbers get increasingly big, approach infinity, notice how this fraction is going to behave. Negative four over one is negative four. But if we had negative four over, say, a100, that would give us negative 0.04. If we had negative 4 over a 1000, that would give us negative 0.004. If we had negative 4 over, say, 10,000, well, that would give us negative 0.0004. You get the idea. And this is how the pattern is going to go on. And notice how the bigger and bigger numbers we plug in, the closer and closer we get to 0. And so because of this, we can see that this entire term right here is is going to go to 0. And this is what happens anytime you have an x in the denominator, but not in the numerator. The entire thing is going to get closer and closer to the value of 0. So So this entire limit is just gonna be equal to 0 +2, which is just simply 2. So the solution to this limit is 2. And that's how you can solve this problem. So I hope you found this video helpful, and let's move on.

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23. Intro to Derivatives & Area Under the Curve

Limits at Infinity

Precalculus

23. Intro to Derivatives & Area Under the Curve

Limits at Infinity

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Multiple Choice

Evaluate the following limit: $\lim_{x \to \infty} - \frac{4}{x} + 2$ .

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Verified step by step guidance

Step 1: Understand the problem statement. We need to evaluate the limit of the expression $-\frac{4}{x} + 2$ as $x$ approaches infinity.

Step 2: Analyze the behavior of each term in the expression as $x$ approaches infinity. The term $-\frac{4}{x}$ will approach 0 because the denominator $x$ becomes very large, making the fraction very small.

Step 3: Consider the constant term $2$. Since it is a constant, it does not change as $x$ approaches infinity.

Step 4: Combine the results from steps 2 and 3. As $x$ approaches infinity, $-\frac{4}{x}$ approaches 0, and the constant $2$ remains unchanged.

Step 5: Conclude that the limit of the expression $-\frac{4}{x} + 2$ as $x$ approaches infinity is simply the constant term $2$.