## Limit at Infinity Problems and Solutions

### Limits at Infinity: Problems and Solutions

Are you working to solve problems about $\displaystyle{\lim_{x \to \infty}}$ and $\displaystyle{\lim_{x \to\, -\infty}}$? Let’s look at common limit at infinity problems and solutions so you can learn to solve them routinely.

CALCULUS SUMMARY: Limits at Infinity

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.

Problem #1: Polynomial

**(a)**Find $\displaystyle{\lim_{x \to \infty} \left(3x^3 + 947x^2 – \sqrt{x} \right)}.$

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

Problem #2: Numerator & denominator have the same highest power

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

Problem #3: Denominator has highest power

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

Problem #4: Numerator has the highest power

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

Problem #5: sin & cos

**(a)**Find $\displaystyle{\lim_{x \to \infty} \sin(x)}$.

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

Limit at Infinity Problems with Square Roots

We have a separate page to help you specifically with Limit at Infinity Problems with Square Roots.

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