Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. $\lim _{x \rightarrow 2}\left(\frac{x}{x+1}+\frac{3}{x-1}\right)=\lim _{x \rightarrow 2} \frac{x}{x+1}+\lim _{x \rightarrow 2} \frac{3}{x-1}$

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. $\lim _{x \rightarrow 1} \frac{x-3}{x^{2}+2 x-4}=\frac{\lim _{x \rightarrow 1}(x-3)}{\lim _{x \rightarrow 1}\left(x^{2}+2 x-4\right)}$

In the following Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

$$\lim _{x \rightarrow 0} \frac{x^2+x+1}{x}=\lim _{x \rightarrow 0} \frac{2 x+1}{1}=1$$

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.

$$\lim _{x \rightarrow 1} \dfrac{x-3}{x^2+2 x-4}=\dfrac{\lim _{x \rightarrow 1}(x-3)}{\lim _{x \rightarrow 1}\left(x^2+2 x-4\right)}$$

Determine whether the statement is true or false. Explain your answer. If y=L is a horizontal asymptote for the curve y=f(x), then it is possible for the graph of f to intersect the line y=L infinitely many times.

Determine whether the statement is true or false.

$$\lim _{x \rightarrow 1} \frac{x-3}{x^2+2 x-4}=\frac{\lim _{x \rightarrow 1}(x-3)}{\lim _{x \rightarrow 1}\left(x^2+2 x-4\right)}$$

Will interpolation between known data points always give an accurate continuous path? If so, explain why. If not, give a counterexample.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If |f| is continuous at a, so is f.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.

If the line x=1 is a vertical asymptote of y=f(x), then f is not defined at 1.

Determine whether the statement is true or false. If it is true,explain why. If it is false, explain why or give an example that disproves the statement $\lim _ { x \rightarrow 1 } \dfrac { x - 3 } { x ^ { 2 } + 2 x - 4 } = \dfrac { \lim _ { x \rightarrow 1 } ( x - 3 ) } { \lim _ { x \rightarrow 1 } \left( x ^ { 2 } + 2 x - 4 \right) }$

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If a function has derivatives from both the right and the left at a point, then it is differentiable at that point.

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. $\lim _{x \rightarrow 1}\left(\frac{2 x}{x-1}-\frac{2}{x-1}\right)=\lim _{x \rightarrow 1} \frac{2 x}{x-1}-\lim _{x \rightarrow 1} \frac{2}{x-1}$

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.**

(a) The slope of the tangent line to the differentiable function at the point $(2,f(2))$ is $\frac{f(2+\Delta x)-f(2)}{\Delta x}$.

Determine whether the statement is true or false. Explain your answer. The velocity of an object represents a change in the object’s position.