91Ó°ÊÓ

Let \(f(x)=\frac{1}{x-1}+\frac{1}{x-4} .\) Show that there is a number \(c \in(1,4)\) such that \(f(c)=0.\)

Evaluate the limits that exist. $$\lim _{x \rightarrow 0} \frac{\sin ^{2} x^{2}}{x^{2}}$$

Determine whether or not the function is continuous at the indicated point. If not, determine whether the discontinuity is a removable discontinuity or an essential discontinuity. If the taller, state whether it is a jump discontinuity, an infinite discontinuity, or neither. $$g(x)=\left\\{\begin{array}{rl} x^{2}+5, & x<2 \\ 10, & x=2 \\ 1+x^{3}, & x>2 \end{array} \quad x=2\right.$$.

Evaluate the limits that exist. $$\lim _{x \rightarrow-1} \frac{x^{2}+1}{3 x^{5}+4}$$

Determine whether or not the function is continuous at the indicated point. If not, determine whether the discontinuity is a removable discontinuity or an essential discontinuity. If the taller, state whether it is a jump discontinuity, an infinite discontinuity, or neither. $$f(x)=\left\\{\begin{array}{cc} \frac{|x-1|}{x-1}, & x \neq 1 \\ 0, & x=1 \end{array} \quad x=1\right.$$.

Decide in the manner of Section 2.1 whether or not the indicated limit exists. Evaluate the limits that do exist. $$\lim _{x \rightarrow 3^{+}} \frac{x+3}{x^{2}-7 x+12}$$

Show that the equation \(x^{3}-4 x+2=0\) has three distinct roots in [-3,3] and locate the roots between consecutive integers.

Evaluate the limits that exist. $$\lim _{x \rightarrow 0}\left(x-\frac{4}{x}\right)$$

Evaluate the limits that exist. $$\lim _{x \rightarrow 0} \frac{\tan ^{2} 3 x}{4 x^{2}}$$

Decide in the manner of Section 2.1 whether or not the indicated limit exists. Evaluate the limits that do exist. $$\lim _{x \rightarrow 1^{-}} \frac{x}{|x|}$$