Evaluate the following limit: lim x → ∞ x + 7 8 x − 100 \lim_{x\rarr\infty}\frac{x+7}{8x-100} lim x → ∞ 8 x − 100 x + 7 .

for this problem, we're asked to evaluate the following limit as x approaches infinity of this function that we have right here. Now recall that we talked about shortcuts that you can use solve situations when you have these kind of rational functions. Now the shortcut was this. If you have x on top, and the degree is higher than the x's on the bottom, then the whole thing is either not going to exist or go to infinity. If you have an x on bottom that with a higher degree than the top, then the entire thing is going to go to 0. However, if you have x of equal degrees on top and bottom, then you're going to get a ratio of the coefficients out in front. Now looking at this example here, I see that the highest degree x we have on top is this one, which is x to the first, and the highest degree x we have on bottom is this one, which is also x to the first. We have equal degrees on top and bottom, so what we need is a ratio of the coefficients out in front, which is going to be 1 over 8. And that right there is the solution to this problem, and that's it. That's how you can use the shortcut to solve these problems very fast. So 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: $\lim_{x\rarr\infty}\frac{x+7}{8x-100}$ .

$7$

$- \frac{1}{100}$

$- \frac{7}{100}$

$\frac{1}{8}$

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Verified step by step guidance

Identify the limit expression: $ \lim_{x \to \infty} \frac{x+7}{8x-100} $.

To evaluate the limit as $ x \to \infty $, divide every term in the numerator and the denominator by $ x $, the highest power of $ x $ in the denominator.

Rewrite the expression: $ \frac{x+7}{8x-100} = \frac{\frac{x}{x} + \frac{7}{x}}{\frac{8x}{x} - \frac{100}{x}} = \frac{1 + \frac{7}{x}}{8 - \frac{100}{x}} $.

As $ x \to \infty $, the terms $ \frac{7}{x} $ and $ \frac{100}{x} $ approach 0.

Thus, the limit simplifies to $ \frac{1 + 0}{8 - 0} = \frac{1}{8} $.