Limits and continuity

Compute $\underset{x\to 3}{\lim }\frac{x^2-5x+6}{x^2-9}$.

Solution available to those who have unlocked the subject.

Compute $\underset{x\to -1}{\lim }\frac{x^3+1}{x+1}$.

Solution available to those who have unlocked the subject.

Compute $\underset{x\to 0}{\lim }\frac{\sqrt{1+x}-1}{x}$.

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Compute $\underset{x\to +\infty }{\lim }\frac{4x^2-3x}{2x^2+7}$.

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Compute $\underset{x\to +\infty }{\lim }\frac{x^3-2}{5x+1}$.

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Compute $\underset{x\to +\infty }{\lim }\left(\sqrt{x^2+6x}-x\right)$.

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Compute $\underset{x\to 0}{\lim }\frac{\ln (1+5x)}{2x}$.

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

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Compute $\underset{x\to +\infty }{\lim }\frac{2^x+x^{10}}{3\cdot 2^x}$.

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Compute the right-hand and left-hand limits of $f(x)=\frac{2x}{3-x}$ as $x\to 3$.

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Find the asymptotes of $f(x)=\frac{3x-1}{x+2}$.

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Find the asymptotes of $f(x)=\frac{2x^2+1}{x-1}$.

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Find $a$ so that the function equal to $a{x}^2$ for $x\le 1$ and to $2x+3$ for $x>1$ is continuous on $\mathbb{R}$.

Solution available to those who have unlocked the subject.

Prove that the equation $x^3-3x+1=0$ has a solution in the interval $(0,1)$.

Solution available to those who have unlocked the subject.