Limits and continuity

Compute

$$\underset{x\to 2}{\lim }\frac{x^2-4}{x^2-3x+2}.$$

Solution available to those who have unlocked the subject.

Let $f(x)=\frac{x^2+1}{x-3}$. Determine whether there exists a neighborhood of $x_0=5$ in which $f$ is positive, and explain which theorem guarantees it.

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Compute

$$\underset{x\to 0}{\lim }\frac{\sin 3x}{\sin 5x}.$$

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Compute

$$\underset{x\to +\infty }{\lim }\left(\sqrt{x^2+2x}-\sqrt{x^2-x}\right).$$

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Compute

$$\underset{x\to 0}{\lim }\frac{1-\cos 2x}{x^2}.$$

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Compute

$$\underset{x\to +\infty }{\lim }{\left(1+\frac{3}{x}\right)}^{2x}.$$

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Compute

$$\underset{x\to 0}{\lim }\frac{\ln (1+4x)}{e^{3x}-1}.$$

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Compute

$$\underset{x\to +\infty }{\lim }\frac{x^3+\ln x}{e^x}.$$

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Compute the right-hand and left-hand limits as $x\to 1$ of

$$f(x)=\frac{x+2}{x^2-1}$$

and determine whether the limit as $x\to 1$ exists.

Solution available to those who have unlocked the subject.

Find the asymptotes of

$$f(x)=\frac{2x^2+1}{x+1}.$$

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Determine $a\in \mathbb{R}$ so that the function

$$f(x)=\{\begin{array}{ll}\frac{\sin 3x}{x}& \text{if }x\ne 0\\ a& \text{if }x=0\end{array}$$

is continuous at $x=0$.

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Classify the point $x=0$ for $f(x)=\frac{|x|}{x}$ and the point $x=1$ for $g(x)=\frac{x^2-1}{x-1}$.

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Prove that the equation

$$x^3-4x+1=0$$

has three real solutions, and give for each one an interval of length $1$ that contains it.

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Compute

$$\underset{x\to 0}{\lim }\frac{e^{x^2}-1}{1-\cos x}.$$

Solution available to those who have unlocked the subject.