If you just need practice with limits problems for now, previous students have found what’s below super-helpful. And if you have questions, please ask on our Forum!

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)** $\displaystyle{\lim_{x \to -\infty} \cos(x)}$

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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