**Update:** We now have *much* more interactive ways for you to learn about the foundational concept of Limits, making heavy use of Desmos graphing calculators so you can work with these ideas for yourself, and develop your problem solving skills step-by-step. Please visit our Limits Chapter to *really* get this material down.

It’s all free, and waiting for you! (Why? Just because we’re educators who believe you deserve the chance to develop a better understanding of Calculus for yourself, and so we’re aiming to provide that. We hope you’ll take advantage!)

If you just need practice with limit-at-infinity problems for now, previous students have found what’s below super-helpful. And if you have questions, please ask on our Forum! It’s also free for your use.

Summary

Here's a summary of our blog post "Limits at Infinity: What You Need to Know." That post goes step-by-step to build up the ideas you need to know to solve these problems.

(Problems where $x \to \infty$ and that involve square roots deserve their own topic: Limit at Infinity Problems with Square Roots.)

(Problems where $x \to \infty$ and that involve square roots deserve their own topic: Limit at Infinity Problems with Square Roots.)

Problem #1: Polynomial 3*x*^3 + ...

Find the requested limits.**(a)** $\displaystyle{\lim_{x \to \infty} \left(3x^3 + 947x^2 - \sqrt{x} \right)}$**(b)** $\displaystyle{\lim_{x \to -\infty} \left(3x^3 + 947x^2 - \sqrt{x} \right)}$

Problem #2: Polynomial *x - x*^2

Find $\displaystyle{\lim_{x \to \infty}\left( x - x^2 \right)}$.

Problem #3: Denominator has highest power

Find $\displaystyle{\lim_{x \to \infty}\frac{4x^3 + 2x -24}{x^4 - x^2 + 84 } }.$

Problem #4: Denominator (again) has the highest power

Find $\displaystyle{\lim_{x \to \infty} \frac{x + 7}{x^3 -x +2}}$.
\begin{array}{lllll} \text{(A) }1 && \text{(B) }0 && \text{(C) }\infty && \text{(D) }\dfrac{7}{2} && \text{(E) none of these} \end{array}

Problem #5: Numerator has the highest power

Find $\displaystyle{\lim_{x \to \infty}\frac{x^3 +2}{3x^2 + 4}}.$

Problem #6: Numerator (again) has the highest power

Find $\displaystyle{\lim_{x \to \infty} \frac{x^2 + 3x}{x+1}}$.
\begin{array}{lllll} \text{(A) }1 && \text{(B) }0 && \text{(C) }\infty && \text{(D) }3 && \text{(E) none of these} \end{array}

Problem #7: Numerator & denominator have the same highest power

Find $\displaystyle{\lim_{x \to \infty}\frac{5x^2 -7}{3x^2 + 8}}.$

Problem #8: Horizontal asymptotes

Find the horizontal asymptotes of $\displaystyle{\frac{5x^2 + x -3}{3x^2 - 2x + 5}}$.

Problem #9: sin & cos

Find the requested limits.**(a)** $\displaystyle{\lim_{x \to \infty} \sin(x)}$**(b)** $\displaystyle{\lim_{x \to -\infty} \cos(x)}$

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Problems where $x \to \infty$ and that involve square roots deserve their own topic: Limit at Infinity Problems with Square Roots.

We'd love to hear:

- What questions do you have about the solutions above?
- Which ones are giving you the most trouble?
- What other Limits problems are you trying to work through for your class?

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