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PROBLEM:

Evaluate $\displaystyle \lim\limits_{x\to 0}\left(\displaystyle \frac{\sin\left(x\right)\sin\left(2x\right)}{1-\cos\left(x\right)}\right).$

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SOLUTION:

Direct substitution of $x=0$ gives the indeterminate form $\frac{0}{0}$. Therefore, we should apply trigonometric identities.

We know that $\sin\left(2x\right)=2\sin\left(x\right)\cos\left(x\right)$, so we can rewrite the original function as

We also know the Pythagorean identity $\sin^2\left(x\right)=1-\cos^2\left(x\right)$. So,

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PROBLEM:

Evaluate $\displaystyle \lim\limits_{x\to 0}\left(\frac{1-\cos^2\left(x\right)}{1+\cos\left(x\right)}\right)$

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SOLUTION:

This problem can be solved using a direct substitution of $x=0$. That is

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PROBLEM:

Evaluate $\displaystyle \lim_{x\to 0}\left(\frac{\sin^3x}{\sin x-\tan x}\right)$.

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SOLUTION:

A straight substitution of $\displaystyle x=\frac{\pi }{4}$ leads to the indeterminate form $\displaystyle \frac{0}{0}$ which is meaningless.

Therefore, to evaluate the limit of the given function, we proceed as follows

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PROBLEM:

Evaluate $\displaystyle \lim\limits_{x\to \frac{\pi }{4}}\left(\frac{\tan\:2x}{\sec\:2x}\right)$.

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SOLUTION:

A straight substitution of $x=\frac{\pi }{4}$ leads to the indeterminate form $\frac{0}{0}$ which is meaningless.

Therefore, to evaluate the limit of the given function, we proceed as follows

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PROBLEM:

Evaluate $\displaystyle \lim\limits_{x\to 3}\left(\frac{\sqrt{x^2-9}}{x-3}\right)$.

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SOLUTION:

A straight substitution of $x=3$ leads to the indeterminate form $\frac{0}{0}$ which is meaningless.

Therefore, to evaluate the limit of the given function, we proceed as follows

Since the function’s limit is different from the left to its limits from the right, the limit does not exist. 

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PROBLEM:

Evaluate $\displaystyle \lim\limits_{x\to 0}\left(\frac{1}{x}\left(\frac{1}{3}-\frac{1}{\sqrt{x+9}}\right)\right)$

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SOLUTION: 

A straight substitution of $x=0$ leads to the indeterminate form $0\cdot 0$ which is meaningless.

Therefore, to evaluate the limit of the given function, we proceed as follows

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Evaluate $\displaystyle \lim\limits_{x\to 3}\left(\frac{x-3}{\sqrt{x-2}-\sqrt{4-x}}\right)$

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Solution:

A straight substitution of $x=3$ leads to the indeterminate form $\frac{0}{0}$ which is meaningless.

Therefore, to evaluate the limit of the given function, we proceed as follows.

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PROBLEM:

Evaluate $\displaystyle \lim\limits_{x\to 2}\left(\frac{x^3-8}{x^2-4\:}\right)$

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SOLUTION:

A straight substitution of  $x=2$ leads to the indeterminate form $\frac{0}{0}$ which is meaningless.

Therefore, to evaluate the limit of the given function, we proceed as follows

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PROBLEM:

Evaluate $\displaystyle \lim\limits_{x\to 4}\left(\frac{\frac{1}{x}-\frac{1}{4}}{x-4\:}\right)$.

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SOLUTION:

A straight substitution of $x=4$ leads to the indeterminate form $\frac{0}{0}$ which is meaningless.

Therefore, to evaluate the limit of the given function, we proceed as follows:

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PROBLEM:

Evaluate $\displaystyle \lim\limits_{x\to 8}\:\frac{\sqrt[3]{x}-2}{x-8}$.

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SOLUTION:

A straight substitution of $x=8$ leads to the indeterminate form $\frac{0}{0}$ which is meaningless.

Therefore, to evaluate the limit of the given function, we proceed as follows

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